#495678

order	512
nilpotency class	3
isoclinism	isoclinic class of #3026
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	C₂×#120
commutator	C₂³	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₈×#120
prime radical	#3025	C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

934 conjugacy classes of subgroups; only the 264 normal subgroups are shown.

normal subgroups only

This group has 934 conjugacy classes of subgroups — too many to draw. Only its 264 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2,6,12,7,5,23,24,8,13,26,36,27,25,11,35)(3,9,17,31,18,15,42,44,16,30,45,56,46,43,29,55)(4,10,21,34,22,19,47,49,20,33,50,58,51,48,32,57)(14,37,40,60,41,61,59,64,39,28,52,54,63,38,53,62);(1,3)(2,9)(4,14)(5,15)(6,17)(7,18)(8,16)(10,28)(11,29)(12,31)(13,30)(19,38)(20,39)(21,40)(22,41)(23,42)(24,44)(25,43)(26,45)(27,46)(32,53)(33,37)(34,54)(35,55)(36,56)(47,59)(48,61)(49,62)(50,52)(51,63)(57,64)(58,60);(1,4)(2,10)(3,14)(5,19)(6,21)(7,22)(8,20)(9,28)(11,32)(12,34)(13,33)(15,38)(16,39)(17,40)(18,41)(23,47)(24,49)(25,48)(26,50)(27,51)(29,53)(30,37)(31,54)(35,57)(36,58)(42,59)(43,61)(44,62)(45,52)(46,63)(55,64)(56,60);(1,5,6,24,7,13,23,36,8,25,26,35,27,2,11,12)(3,15,17,44,18,30,42,56,16,43,45,55,46,9,29,31)(4,19,21,49,22,33,47,58,20,48,50,57,51,10,32,34)(14,38,40,62,41,37,59,60,39,61,52,64,63,28,53,54)

Polycyclic presentation:

2^9|a^2=g,d^2=g,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=di,f^b=fi,e^c=ei

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72342018
maximal subgroup of	#72224962, #72342018, #72342023, …
maximal normal subgroup of	#72224962, #72342018, #72342023, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402874);