#93522

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×Q₈	C₂²
prime radical	C₂⁴	#95
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×C₄×Q₈ ▷ C₂⁴ ▷ C₂ ▷ C₁

530 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 530 conjugacy classes of subgroups — too many to draw. Only its 84 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) = 256

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

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

Polycyclic presentation:

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

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, 718);