#99285

order	512
nilpotency class	3
isoclinism	isoclinic class of #99283
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#382
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

222 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=e,c^2=dg,d^2=h,e^2=g,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,c^b=ci,f^b=fgh,f^c=fgh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6481);