#6497474

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	C₂×#121	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1678 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)(5,6,9,12)(7,10,13,17)(8,15,16,11)(14,20,19,18)(21,22)(23,28)(24,36)(25,30)(26,40)(27,31)(29,38)(32,45)(33,42)(34,43)(35,49)(37,44)(39,50)(41,48)(46,51)(47,52);(2,4)(5,7)(6,10)(8,14)(9,13)(11,18)(12,17)(15,20)(16,19)(21,23)(22,28)(24,32)(25,33)(26,35)(27,34)(29,41)(30,42)(31,43)(36,45)(37,46)(38,48)(39,47)(40,49)(44,51)(50,52);(7,13)(10,17)(14,19)(18,20)(21,24,25,37,27,39,26,38)(22,29,40,50,31,44,30,36)(23,32,33,46,34,47,35,48)(28,41,49,52,43,51,42,45);(7,13)(10,17)(14,19)(18,20)(21,25,27,26)(22,30,31,40)(23,33,34,35)(24,37,39,38)(28,42,43,49)(29,36,44,50)(32,46,47,48)(41,45,51,52);(5,8,9,16)(6,11,12,15)(7,14,13,19)(10,18,17,20)(23,34)(28,43)(32,47)(33,35)(41,51)(42,49)(45,52)(46,48)

Polycyclic presentation:

2^9|a^2=h,c^2=gi,d^2=i,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,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, 6404670);