#124464

order	512
nilpotency class	4
isoclinism	isoclinic class of #11769
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1505
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×#97	C₂³
socle	C₂	#7587
prime radical	#3147	C₄
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

347 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 347 conjugacy classes of subgroups — too many to draw. Only its 92 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 — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=e,c^2=fgh,d^2=gi,e^2=h,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=cf,d^a=df,g^a=gi,d^b=dg,e^b=efgi,f^b=fi,g^b=gi,d^c=di,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 31660);