#93598

order	512
nilpotency class	4
isoclinism	isoclinic class of #93583
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1296
commutator	C₂×C₄²	C₄²
Frattini, cosocle	C₄²×C₈	C₂²
prime radical	C₂³	#343
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₄²×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

190 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 74 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:

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

Polycyclic presentation:

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

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, 794);