#6484659

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470860
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1672 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)(5,6)(7,10)(8,11)(9,12)(13,17)(14,18)(15,19)(16,20)(21,22,27,32)(23,33,35,44)(24,38,41,45)(25,31,26,30)(28,37,43,36)(29,39,46,40)(34,42,48,52)(47,49,51,50);(2,4)(5,7)(6,10)(8,15)(9,13)(11,19)(12,17)(14,16)(18,20)(21,23,26,37,27,35,25,36)(22,28,31,44,32,43,30,33)(24,34,40,50,41,48,39,49)(29,42,38,51,46,52,45,47);(7,13)(10,17)(14,15)(18,19)(21,24,25,39,27,41,26,40)(22,29,30,45,32,46,31,38)(23,34,36,49,35,48,37,50)(28,42,33,47,43,52,44,51);(5,8,9,16)(6,11,12,20)(7,14,13,15)(10,18,17,19)(23,35)(28,43)(33,44)(34,48)(36,37)(42,52)(47,51)(49,50);(6,12)(10,17)(11,20)(18,19)(23,35)(28,43)(33,44)(34,48)(36,37)(42,52)(47,51)(49,50)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gi,d^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dhi,e^b=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, 6391855);