#94772

order	512
nilpotency class	6
isoclinism	isoclinic class of #94771
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#7481
commutator	#125	C₄²
Frattini, cosocle	#3138	C₂²
prime radical	C₂×#127	D₄
LCS	G ▷ #125 ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #3138 ◁ G
LECS	G ▷ #3138 ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

176 conjugacy classes of subgroups; only the 34 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 34 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) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 2 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=hi,e^2=i,f^2=hi,g^2=i,b^a=bc,c^a=cf,e^a=ehi,f^a=fg,g^a=gh,h^a=hi,d^b=df,f^b=fh,g^b=gh,d^c=dgh,f^c=fh,g^c=gi,e^d=ei,f^d=fhi,g^d=gi

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1968);