#6565117

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
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	#2535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1367 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)(3,7)(4,6)(5,8)(9,10,13,16)(11,14,17,20)(12,15,19,22)(18,21,23,24)(25,26,30,35)(27,36,38,32)(28,40,42,49)(29,34,31,44)(33,45,50,41)(37,52,53,56)(39,48,43,47)(46,55,51,54);(1,3,5,4)(2,6,8,7)(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24)(25,27)(26,32)(28,37)(29,43)(30,38)(31,39)(33,46)(34,48)(35,36)(40,56)(41,55)(42,53)(44,47)(45,54)(49,52)(50,51);(11,17)(14,20)(18,23)(21,24)(25,28,30,42)(26,33,35,50)(27,37,38,53)(29,41,31,45)(32,46,36,51)(34,49,44,40)(39,54,43,55)(47,56,48,52);(1,3,5,4)(2,7,8,6)(9,12)(10,15)(11,18)(13,19)(14,21)(16,22)(17,23)(20,24)(27,38)(32,36)(37,53)(39,43)(46,51)(47,48)(52,56)(54,55);(1,4,5,3)(2,6,8,7)(12,19)(15,22)(18,23)(21,24)(25,29,30,31)(26,34,35,44)(27,39,38,43)(28,41,42,45)(32,47,36,48)(33,49,50,40)(37,54,53,55)(46,56,51,52)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6472313);