#6529404

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500060
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	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2734 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)(21,22,26,31,27,32,25,30)(23,33,36,44,37,43,35,28)(24,29,40,47,39,46,41,48)(34,49,50,45,38,52,51,42);(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,23,27,37)(22,28,32,44)(24,38,39,34)(25,36,26,35)(29,45,46,42)(30,33,31,43)(40,50,41,51)(47,49,48,52);(2,4)(5,7,9,15)(6,10,12,13)(8,16,17,14)(11,19,20,18)(21,24,27,39)(22,29,32,46)(23,34,37,38)(25,41,26,40)(28,42,44,45)(30,48,31,47)(33,49,43,52)(35,51,36,50);(5,8,9,17)(6,11,12,20)(7,14,15,16)(10,18,13,19)(24,39)(29,46)(34,38)(40,41)(42,45)(47,48)(49,52)(50,51);(6,12)(10,13)(11,20)(18,19)(21,25,27,26)(22,30,32,31)(23,35,37,36)(24,40,39,41)(28,43,44,33)(29,47,46,48)(34,50,38,51)(42,49,45,52)

Polycyclic presentation:

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

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, 6436600);