#6798507

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682440
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1334 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 56 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fg,d^2=f,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=di,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6705703);