#125498

order	512
nilpotency class	4
isoclinism	isoclinic class of #125496
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1519
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×#97	C₂³
prime radical	C₂⁴	#102
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

299 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 299 conjugacy classes of subgroups — too many to draw. Only its 72 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) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

(1,2,4,6,5,3,7,8)(9,10,13,19,15,21,14,20)(11,17,23,35,24,37,29,36)(12,25,27,38,28,31,30,18)(16,26,32,22,34,39,33,40);(1,3)(2,5)(4,6)(7,8)(9,11,15,24)(10,16,21,34)(12,26,28,39)(13,23,14,29)(17,25,37,31)(18,36,38,35)(19,33,20,32)(22,30,40,27);(2,3)(6,8)(9,11)(10,17)(12,22)(13,29)(14,23)(15,24)(16,31)(18,33)(19,36)(20,35)(21,37)(25,34)(26,27)(28,40)(30,39)(32,38)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 32694);