#124459

order	512
nilpotency class	4
isoclinism	isoclinic class of #11912
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1530
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₂×#97	C₂³
prime radical	C₂³	C₂×#100
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

348 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 94 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) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 72 points:

(1,2)(3,5)(4,6)(7,8)(9,10,13,20,15,19,38,48,16,22,36,52,14,21,37,51)(11,23,25,54,27,43,34,64,28,42,60,49,26,17,39,41)(12,29,31,61,33,18,44,45,30,47,35,66,32,46,63,50)(24,40,56,68,58,53,62,72,55,69,59,70,57,67,65,71);(2,5)(3,8)(6,7)(9,11,16,28)(10,17,22,43)(12,24,30,55)(13,34,36,39)(14,27,15,26)(18,40,46,69)(19,42,21,23)(20,49,52,54)(25,37,60,38)(29,53,47,67)(31,62,35,65)(32,58,33,57)(41,48,64,51)(44,56,63,59)(45,71,50,72)(61,68,66,70);(2,6)(5,7)(9,12)(10,18,22,46)(11,24)(13,35)(14,32)(15,33)(16,30)(17,40,43,69)(19,47,21,29)(20,50,52,45)(23,53,42,67)(25,59)(26,57)(27,58)(28,55)(31,36)(34,65)(37,44)(38,63)(39,62)(41,70,64,68)(48,61,51,66)(49,72,54,71)(56,60)

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=fgh,d^2=g,e^2=h,f^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,h^b=hi,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, 31655);