#6565445

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535815
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×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1316 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)(27,35,38,49)(28,40,41,32)(29,43,36,33)(31,44,46,39)(37,51,52,48)(42,47,50,55)(45,53,54,56);(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24)(25,27,36,44,30,38,29,39)(26,31,43,49,34,46,33,35)(28,37,50,56,41,52,42,53)(32,45,55,51,40,54,47,48);(2,5)(6,8)(11,17)(14,20)(18,23)(21,24)(25,28)(26,32)(27,37)(29,42)(30,41)(31,45)(33,47)(34,40)(35,48)(36,50)(38,52)(39,53)(43,55)(44,56)(46,54)(49,51);(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)(31,46)(35,49)(37,52)(39,44)(45,54)(48,51)(53,56);(12,19)(15,22)(18,23)(21,24)(27,38)(28,41)(31,46)(32,40)(35,49)(39,44)(42,50)(47,55)

Polycyclic presentation:

2^9|a^2=hi,b^2=fi,d^2=gi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,e^b=ei,e^c=ei,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, 6472641);