#6564698

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×#547	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1781 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)(5,6,9,12)(7,10,13,16)(8,11,15,18)(14,17,19,20)(21,22,27,31)(23,28,35,44)(24,36,38,45)(25,30,26,39)(29,40,46,37)(32,47,50,51)(33,42,34,43)(41,48,52,49);(1,2,3,4)(5,7)(6,10)(8,14)(9,13)(11,17)(12,16)(15,19)(18,20)(21,23,27,35)(22,28,31,44)(24,32,38,50)(25,34,26,33)(29,41,46,52)(30,43,39,42)(36,47,45,51)(37,49,40,48);(7,13)(10,16)(14,19)(17,20)(21,24)(22,29)(23,32)(25,37)(26,40)(27,38)(28,41)(30,45)(31,46)(33,48)(34,49)(35,50)(36,39)(42,47)(43,51)(44,52);(5,8)(6,11)(7,14)(9,15)(10,17)(12,18)(13,19)(16,20)(21,25,27,26)(22,30,31,39)(23,33,35,34)(24,37,38,40)(28,42,44,43)(29,45,46,36)(32,48,50,49)(41,47,52,51);(8,15)(11,18)(14,19)(17,20)(21,25,27,26)(22,30,31,39)(23,34,35,33)(24,37,38,40)(28,43,44,42)(29,45,46,36)(32,49,50,48)(41,51,52,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=fi,d^2=i,e^2=i,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, 6471894);