#6482631

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470185
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)(3,6)(4,5)(7,8)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,23)(20,24)(25,26,29,34)(27,36,38,31)(28,33,40,49)(30,43,42,35)(32,44,48,41)(37,51,53,46)(39,47,55,52)(45,54,50,56);(2,5)(6,8)(9,11)(10,14)(12,19)(13,17)(15,23)(16,21)(18,20)(22,24)(25,27,29,38)(26,31,34,36)(28,37,40,53)(30,39,42,55)(32,45,48,50)(33,46,49,51)(35,47,43,52)(41,54,44,56);(11,17)(14,21)(18,19)(22,23)(25,28,29,40)(26,32,34,48)(27,37,38,53)(30,44,42,41)(31,45,36,50)(33,35,49,43)(39,56,55,54)(46,47,51,52);(9,12,13,20)(10,15,16,24)(11,18,17,19)(14,22,21,23)(25,28)(26,33)(27,37)(29,40)(30,41)(31,46)(32,43)(34,49)(35,48)(36,51)(38,53)(39,54)(42,44)(45,52)(47,50)(55,56);(1,3,4,7)(2,6,5,8)(10,16)(14,21)(15,24)(22,23)(28,40)(32,48)(33,49)(37,53)(41,44)(45,50)(46,51)(54,56)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=h,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,d^c=di,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6389827);