#124496

order	512
nilpotency class	4
isoclinism	isoclinic class of #11912
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂×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

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

normal subgroups only

This group has 278 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=g,c^2=fgh,d^2=gi,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=efg,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, 31692);