#99284

order	512
nilpotency class	3
isoclinism	isoclinic class of #99283
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

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

Polycyclic presentation:

2^9|a^2=f,b^2=eg,c^2=d,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, 6480);