#6566385

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1689 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:

(2,3)(5,6,9,12)(7,10,13,16)(8,11,15,18)(14,17,19,20)(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);(1,2,4,3)(5,7)(6,10)(8,14)(9,13)(11,17)(12,16)(15,19)(18,20)(21,23,27,33)(22,28,31,42)(24,32,38,47)(25,34,26,35)(29,41,45,51)(30,43,39,49)(36,46,44,52)(37,48,40,50);(1,3,4,2)(7,13)(10,16)(14,19)(17,20)(21,24)(22,29)(23,32)(25,37)(26,40)(27,38)(28,41)(30,44)(31,45)(33,47)(34,48)(35,50)(36,39)(42,51)(43,52)(46,49);(5,8)(6,11)(7,14)(9,15)(10,17)(12,18)(13,19)(16,20)(23,33)(28,42)(32,47)(34,35)(41,51)(43,49)(46,52)(48,50);(8,15)(11,18)(14,19)(17,20)(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=fi,c^2=f,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, 6473581);