#93511

order	512
nilpotency class	4
isoclinism	isoclinic class of #93511
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂²×C₄

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

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

normal subgroups only

This group has 463 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)(3,14,16,44)(4,20,23,52)(6,13,28,40)(7,11,27,41)(8,29,30,48)(9,33,35,24)(12,42,39,22)(15,31,46,38)(17,51,49,61)(18,34,50,60)(19,45,26,57)(21,43,53,64)(25,37,56,62)(32,55,59,47)(36,58,63,54);(1,3)(2,8,11,31)(4,21)(5,26,27,16)(6,17)(7,19)(9,34,36,51)(10,38)(12,32,13,43)(14,20,45,37)(15,24)(18,22)(23,55,56,53)(25,47)(28,50,42,49)(29,58,46,33)(30,41)(35,61)(39,64)(40,59)(44,52)(48,54)(57,62)(60,63)

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fhi,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, 707);