#125557

order	512
nilpotency class	4
isoclinism	isoclinic class of #125534
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₄

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

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

normal subgroups only

This group has 243 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=g,c^2=fgi,d^2=gi,e^2=h,f^2=i,h^2=i,b^a=bd,c^a=cfg,d^a=df,g^a=gi,c^b=ch,d^b=dg,e^b=efg,f^b=fi,h^b=hi,d^c=di,e^d=ei

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 32753);