#6510997

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500535
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

1271 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)(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23)(25,26)(27,36)(28,33)(29,34)(30,45)(31,35)(32,40)(37,52)(38,53)(39,46)(41,56)(42,49)(43,51)(44,50)(47,54)(48,55);(2,5)(6,8)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(27,37)(32,46)(36,52)(38,41)(39,40)(47,48)(53,56)(54,55);(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(25,27,29,39,31,37,30,40)(26,32,45,52,35,46,34,36)(28,41,44,54,43,38,42,55)(33,48,49,53,51,47,50,56);(1,3,4,7)(2,6,5,8)(12,20)(15,24)(17,19)(21,23)(25,28,31,43)(26,33,35,51)(27,38,37,41)(29,44,30,42)(32,47,46,48)(34,50,45,49)(36,53,52,56)(39,55,40,54);(11,18)(14,22)(17,19)(21,23)(25,29,31,30)(26,34,35,45)(27,39,37,40)(28,42,43,44)(32,36,46,52)(33,49,51,50)(38,54,41,55)(47,56,48,53)

Polycyclic presentation:

2^9|a^2=hi,c^2=gi,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=ehi,d^c=dh,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, 6418193);