#508318

order	512
nilpotency class	4
isoclinism	isoclinic class of #508316
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#29732
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#32948	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #3573 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

672 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 672 conjugacy classes of subgroups — too many to draw. Only its 132 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:

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

Polycyclic presentation:

2^9|a^2=fi,d^2=h,e^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,c^b=cg,e^b=ei,f^b=fi,d^c=de,e^c=ei,f^c=fh,h^c=hi,g^d=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 415514);