#93552

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1297
commutator	C₂²×C₈	C₄²
Frattini, cosocle	C₂×#405	C₂²
socle	C₂	#7036
prime radical	C₂⁴	#104
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄² ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×#405 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

254 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 56 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)(3,14,16,43,17,44,48,59)(4,9,22,33)(6,13,27,39)(7,11,26,40)(8,28,29,20,30,51,58,47)(12,41,38,21)(15,42,45,63,46,31,57,37)(18,50,49,64,19,32,25,55)(23,36,54,61)(24,34,53,62)(35,56,60,52);(1,3)(2,8,11,30)(4,20)(5,25,26,49)(6,19)(7,17)(9,32,34,50)(10,37)(12,42,13,31)(14,36,44,35)(15,23)(16,27,48,41)(18,21)(22,45,53,57)(24,47)(28,56,51,54)(29,38)(33,59)(39,58)(40,63)(43,62)(46,52)(55,60)(61,64)

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fhi,d^2=f,g^2=i,h^2=i,b^a=bc,c^a=cf,e^a=efghi,g^a=gi,c^b=cg,d^b=dh,f^b=fi,g^b=gi,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, 748);