#6529105

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499804
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#121	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2798 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6436301);