#6487258

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470894
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	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1229 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)(11,14)(12,15)(13,16)(17,21)(18,22)(19,23)(20,24)(25,26,32,36)(27,39,43,51)(28,44,46,53)(29,35,45,52)(30,38,31,37)(33,42,50,41)(34,47,54,40)(48,56,49,55);(1,3,5,4)(2,6,7,8)(9,11)(10,14)(12,19)(13,17)(15,23)(16,21)(18,20)(22,24)(25,27,31,42,32,43,30,41)(26,33,38,51,36,50,37,39)(28,45,40,48,46,29,47,49)(34,52,44,55,54,35,53,56);(11,17)(14,21)(18,19)(22,23)(25,28,30,47,32,46,31,40)(26,34,37,53,36,54,38,44)(27,29,41,48,43,45,42,49)(33,35,39,55,50,52,51,56);(9,12,13,20)(10,15,16,24)(11,18,17,19)(14,22,21,23)(25,29)(26,35)(27,40)(28,41)(30,48)(31,49)(32,45)(33,44)(34,39)(36,52)(37,55)(38,56)(42,46)(43,47)(50,53)(51,54);(1,4,5,3)(2,6,7,8)(10,16)(14,21)(15,24)(22,23)(26,36)(28,46)(29,45)(33,50)(37,38)(39,51)(40,47)(48,49)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6394454);