#93513

order	512
nilpotency class	4
isoclinism	isoclinic class of #93511
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	C₂²×#131	C₂²
socle	C₂	#7035
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×#131 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 447 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) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 2 generators on 64 points:

(1,2,5,11,8,14,29,44)(3,15,17,45,20,47,27,58)(4,21,23,53,26,39,56,63)(6,13,30,42,7,12,28,43)(9,31,32,16,35,48,41,50)(10,36,38,24,40,57,64,25)(18,37,52,61,19,46,51,62)(22,33,54,60,49,34,55,59);(1,3,6,18,8,20,7,19)(2,9,12,33,14,35,13,34)(4,22,24,50,26,49,25,16)(5,27,28,52,29,17,30,51)(10,37,39,15,40,46,21,47)(11,41,42,60,44,32,43,59)(23,55,36,31,56,54,57,48)(38,62,53,45,64,61,63,58)

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,f^a=fi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,e^d=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 709);