#94602

order	512
nilpotency class	6
isoclinism	isoclinic class of #7523
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1404
commutator	#125	C₂×C₈
Frattini, cosocle	#1995	C₂²
socle	C₂	#7376
prime radical	C₂³	#352
LCS	G ▷ #125 ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1995 ◁ G
LECS	G ▷ #1995 ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

121 conjugacy classes of subgroups; only the 26 normal subgroups are shown.

normal subgroups only

This group has 121 conjugacy classes of subgroups — too many to draw. Only its 26 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=hi,d^2=f,e^2=hi,f^2=i,g^2=i,b^a=bc,c^a=ce,e^a=eg,g^a=gh,h^a=hi,c^b=cgh,d^b=de,g^b=ghi,d^c=dgh,e^c=eh,g^c=gi,e^d=ehi,g^d=gi

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1798);