#6528723

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499804
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

1339 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)(11,14,17,20)(12,15,19,22)(18,21,23,24)(25,26,30,34)(27,35,38,50)(28,32,40,48)(29,42,44,33)(31,43,46,37)(36,51,39,53)(41,56,55,49)(45,54,47,52);(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24)(25,27,30,38)(26,31,34,46)(28,39,40,36)(29,43,44,37)(32,47,48,45)(33,50,42,35)(41,52,55,54)(49,51,56,53);(2,5)(6,8)(11,17)(12,19)(14,20)(15,22)(25,28)(26,32)(27,36)(29,41)(30,40)(31,45)(33,49)(34,48)(35,51)(37,52)(38,39)(42,56)(43,54)(44,55)(46,47)(50,53);(9,12,13,19)(10,15,16,22)(11,18,17,23)(14,21,20,24)(28,40)(32,48)(36,39)(41,55)(45,47)(49,56)(51,53)(52,54);(1,3,4,7)(2,6,5,8)(11,17)(14,20)(18,23)(21,24)(28,40)(32,48)(36,39)(41,55)(45,47)(49,56)(51,53)(52,54)

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=hi,e^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=ehi,f^b=fi,d^c=dh,e^c=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, 6435919);