#125461

order	512
nilpotency class	4
isoclinism	isoclinic class of #125422
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

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

246 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 64 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=fh,d^2=gi,e^2=h,g^2=i,h^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fi,c^b=ch,d^b=dg,e^b=efg,g^b=gi,h^b=hi,d^c=di,e^c=ei,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, 32657);