#125006

order	512
nilpotency class	4
isoclinism	isoclinic class of #124940
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

330 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 76 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:

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

Polycyclic presentation:

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

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, 32202);