#6565229

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535655
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	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1332 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 128

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

Polycyclic presentation:

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

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, 6472425);