#6565869

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1699 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 52 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=i,d^2=h,e^2=hi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,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, 6473065);