#6564714

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
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

1778 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)(23,28)(24,37)(25,30)(26,31)(27,32)(29,38)(33,47)(34,42)(35,43)(36,44)(39,45)(40,46)(41,49)(48,51)(50,52);(1,2,3,4)(5,7)(6,10)(8,14)(9,13)(11,17)(12,16)(15,19)(18,20)(21,23,27,36)(22,28,32,44)(24,33,39,50)(25,35,26,34)(29,41,46,51)(30,43,31,42)(37,47,45,52)(38,49,40,48);(7,13)(10,16)(14,19)(17,20)(21,24)(22,29)(23,33)(25,38)(26,40)(27,39)(28,41)(30,45)(31,37)(32,46)(34,48)(35,49)(36,50)(42,47)(43,52)(44,51);(5,8)(6,11)(7,14)(9,15)(10,17)(12,18)(13,19)(16,20)(21,25,27,26)(22,30,32,31)(23,34,36,35)(24,38,39,40)(28,42,44,43)(29,45,46,37)(33,48,50,49)(41,47,51,52);(8,15)(11,18)(14,19)(17,20)(21,25,27,26)(22,31,32,30)(23,35,36,34)(24,38,39,40)(28,42,44,43)(29,37,46,45)(33,49,50,48)(41,47,51,52)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6471910);