#93549

order	512
nilpotency class	4
isoclinism	isoclinic class of #93526
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

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

normal subgroups only

This group has 257 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)(3,14,16,42,17,43,47,64)(4,9,21,33)(6,13,26,39)(7,11,25,40)(8,27,28,49,29,57,58,19)(12,41,38,20)(15,30,45,59,46,31,51,37)(18,44,24,54,48,32,55,60)(22,36,53,61)(23,35,52,63)(34,56,62,50);(1,3)(2,8,11,29)(4,19,22,49)(5,24,25,55)(6,18)(7,17)(9,32)(10,37)(12,30,13,31)(14,35)(15,23,46,50)(16,26,47,41)(20,48)(21,51)(27,56)(28,38)(33,42,61,64)(34,43)(36,44)(39,58)(40,59)(45,53)(52,57)(54,62,60,63)

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,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, 745);