#94805

order	512
nilpotency class	6
isoclinism	isoclinic class of #7527
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1408
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

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

normal subgroups only

This group has 199 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)(3,13,15,26)(5,11,17,28)(6,12,19,29)(7,22,23,14)(8,18,25,16)(10,20,27,21)(24,31,32,30);(2,7,10,8,12,13,11,24)(3,14)(4,17,19,20)(9,26,27,32,29,23,28,25)(15,31,22,18)(16,30)

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=ghi,e^2=hi,f^2=hi,g^2=i,b^a=bc,c^a=cf,e^a=egh,f^a=fg,g^a=gh,h^a=hi,c^b=ci,d^b=dfgi,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, 2001);