#125032

order	512
nilpotency class	4
isoclinism	isoclinic class of #124937
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1533
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#97	C₂³
prime radical	C₂³	C₂×#100
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₄×#97 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

273 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 273 conjugacy classes of subgroups — too many to draw. Only its 68 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=fh,d^2=g,e^2=h,f^2=i,h^2=i,b^a=bd,c^a=cfgh,d^a=df,g^a=gi,d^b=dg,e^b=efgi,f^b=fi,h^b=hi,e^c=ei,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, 32228);