#93641

order	512
nilpotency class	4
isoclinism	isoclinic class of #93520
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#342
commutator	C₂³×C₄	C₄²
Frattini, cosocle	C₂²×C₄×D₄	C₂²
prime radical	C₂³×D₄	Q₈
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄×D₄ ▷ C₂⁴ ▷ C₂ ▷ C₁

600 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 600 conjugacy classes of subgroups — too many to draw. Only its 68 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fh,d^2=fh,e^2=fh,b^a=bc,c^a=cf,e^a=efgh,f^a=fi,h^a=hi,c^b=cg,d^b=dh,d^c=di

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 837);