#94774

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	#3139	C₂²
prime radical	C₂³	#352
LCS	G ▷ #125 ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #3139 ◁ G
LECS	G ▷ #3139 ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 157 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=hi,d^2=i,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, 1970);