#6565434

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

1314 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)(27,36)(28,40)(29,34)(30,45)(31,35)(32,39)(33,42)(37,52)(38,48)(41,50)(43,47)(44,49)(46,54)(51,53)(55,56);(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24)(25,27,30,39,31,38,29,43)(26,32,34,36,35,47,45,48)(28,37,41,55,42,53,44,54)(33,46,50,51,40,56,49,52);(2,5)(6,8)(11,17)(14,20)(18,23)(21,24)(25,28)(26,33)(27,37)(29,44)(30,41)(31,42)(32,46)(34,50)(35,40)(36,51)(38,53)(39,55)(43,54)(45,49)(47,56)(48,52);(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)(32,47)(36,48)(37,53)(39,43)(46,56)(51,52)(54,55);(12,19)(15,22)(18,23)(21,24)(25,29,31,30)(26,34,35,45)(27,39,38,43)(28,41,42,44)(32,48,47,36)(33,49,40,50)(37,54,53,55)(46,51,56,52)

Polycyclic presentation:

2^9|a^2=h,b^2=f,d^2=gi,e^2=i,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, 6472630);