#124630

order	512
nilpotency class	4
isoclinism	isoclinic class of #11992
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1555
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	C₂×#100
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₄×#97 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

340 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 80 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)(9,10,12,19,14,20,29,36,15,17,27,33,13,18,28,34)(11,21,22,37,24,16,26,31,25,32,38,35,23,30,39,40);(9,11,15,25)(10,16,17,30)(12,26,27,39)(13,24,14,23)(18,32,20,21)(19,35,33,37)(22,28,38,29)(31,34,40,36);(1,3,5,4)(2,6,8,7)(10,17)(12,27)(16,30)(18,20)(21,32)(22,38)(26,39)(28,29)

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=fgh,d^2=g,e^2=h,f^2=i,h^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fi,g^a=gi,d^b=dg,e^b=efgi,f^b=fi,h^b=hi,e^c=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, 31826);