#6891398

order	512
nilpotency class	3
isoclinism	isoclinic class of #6885065
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

1391 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6798594);