#93531

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

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

normal subgroups only

This group has 455 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fhi,d^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, 727);