#94803

order	512
nilpotency class	6
isoclinism	isoclinic class of #7533
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1409
commutator	C₄×C₈	C₄²
Frattini, prime radical, cosocle	#3147	C₂²
socle	C₂	#7498
LCS	G ▷ C₄×C₈ ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #3147 ◁ G
LECS	G ▷ #3147 ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

197 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 197 conjugacy classes of subgroups — too many to draw. Only its 36 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)(3,7,15,21)(5,10,20,25)(6,12,19,27)(8,18,24,16)(11,17,26,14)(13,23,28,32)(22,30,31,29);(2,7,10,8,12,23,11,22)(3,13)(4,17,19,20)(9,21,25,24,27,32,26,31)(15,18,28,30)(16,29)

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=gh,e^2=hi,f^2=hi,g^2=i,b^a=bc,c^a=cf,e^a=eghi,f^a=fg,g^a=gh,h^a=hi,c^b=ci,d^b=dfg,d^c=dgh,e^d=eh,f^d=fhi,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, 1999);