#124289

order	512
nilpotency class	4
isoclinism	isoclinic class of #11867
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

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

353 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 353 conjugacy classes of subgroups — too many to draw. Only its 100 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=fi,c^2=i,d^2=g,e^2=h,g^2=i,h^2=i,b^a=bd,c^a=cf,d^a=df,f^a=fi,d^b=dg,e^b=efgi,g^b=gi,h^b=hi,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, 31485);