#93528

order	512
nilpotency class	4
isoclinism	isoclinic class of #93526
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1297
commutator	C₂²×C₈	C₄²
Frattini, cosocle	C₂³×M₁₆	C₂²
socle	C₂	#7035
prime radical	C₂⁵	Q₁₆
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ C₂³×M₁₆ ◁ G
LECS	G ▷ C₂³×M₁₆ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

482 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 70 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227805, #72227812
maximal normal subgroup of	#72227805, #72227812

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 724);