#6496252

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470161
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

2822 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)(23,33)(24,38)(25,32)(26,31)(27,30)(28,36)(29,40)(34,48)(35,44)(37,43)(39,45)(41,46)(42,49)(47,51)(50,52);(2,4)(5,7)(6,10)(8,14)(9,13)(11,18)(12,17)(15,20)(16,19)(21,23,27,35)(22,28,30,43)(24,34,41,49)(25,37,26,36)(29,42,45,48)(31,44,32,33)(38,52,46,47)(39,51,40,50);(7,13)(10,17)(14,19)(18,20)(21,24,27,41)(22,29,30,45)(23,34,35,49)(25,40,26,39)(28,42,43,48)(31,38,32,46)(33,47,44,52)(36,51,37,50);(7,13)(10,17)(14,19)(18,20)(22,30)(23,35)(29,45)(31,32)(34,49)(36,37)(38,46)(50,51);(5,8,9,16)(6,11,12,15)(7,14,13,19)(10,18,17,20)(23,35)(28,43)(33,44)(34,49)(36,37)(42,48)(47,52)(50,51)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6403448);