#124908

order	512
nilpotency class	4
isoclinism	isoclinic class of #124850
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#410
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#97	C₂³
socle	C₂²	#1466
prime radical	#2397	C₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #1726 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

377 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 377 conjugacy classes of subgroups — too many to draw. Only its 72 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=e,c^2=fgh,d^2=g,e^2=h,h^2=i,b^a=bd,c^a=cfh,d^a=df,f^a=fi,c^b=ci,d^b=dg,e^b=efg,e^d=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, 32104);