#124851

order	512
nilpotency class	4
isoclinism	isoclinic class of #124851
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

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	C₂²×D₈	C₂×C₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

314 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 70 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:

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

Polycyclic presentation:

2^9|a^2=e,c^2=f,d^2=gi,e^2=h,g^2=i,h^2=i,b^a=bd,c^a=cfh,d^a=df,f^a=fi,d^b=dg,e^b=efg,g^b=gi,h^b=hi,d^c=di,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, 32047);