#124443

order	512
nilpotency class	4
isoclinism	isoclinic class of #11867
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

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

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

normal subgroups only

This group has 370 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) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=gh,d^2=gi,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=efg,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, 31639);