#6482393

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470137
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 ◁ #2293 ◁ 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)(25,26,29,33)(27,35,37,31)(28,39,41,50)(30,44,43,34)(32,45,49,42)(36,52,40,56)(38,47,54,51)(46,53,48,55);(2,5)(6,8)(9,11)(10,14)(12,19)(13,17)(15,23)(16,21)(18,20)(22,24)(25,27)(26,31)(28,40)(29,37)(30,38)(32,48)(33,35)(34,47)(36,41)(39,52)(42,55)(43,54)(44,51)(45,53)(46,49)(50,56);(11,17)(14,21)(18,19)(22,23)(25,28,29,41)(26,32,33,49)(27,36,37,40)(30,45,43,42)(31,46,35,48)(34,50,44,39)(38,55,54,53)(47,52,51,56);(9,12,13,20)(10,15,16,24)(11,18,17,19)(14,22,21,23)(27,37)(31,35)(36,40)(38,54)(46,48)(47,51)(52,56)(53,55);(1,3,4,7)(2,6,5,8)(10,16)(14,21)(15,24)(22,23)(26,33)(27,37)(32,49)(34,44)(36,40)(38,54)(39,50)(53,55)

Polycyclic presentation:

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

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, 6389589);