#93532

order	512
nilpotency class	4
isoclinism	isoclinic class of #93529
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1284
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×C₄ ◁ 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:

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

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,h^b=hi,e^c=ei,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, 728);