#124371

order	512
nilpotency class	4
isoclinism	isoclinic class of #11769
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1505
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×#97	C₂³
prime radical	C₂³	#418
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 285 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=fg,d^2=gi,e^2=h,f^2=i,g^2=i,b^a=bd,c^a=cf,d^a=df,g^a=gi,d^b=dg,e^b=efgi,f^b=fi,g^b=gi,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, 31567);