#6622991

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574298
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

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 ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1442 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fgi,d^2=fh,e^2=hi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=ehi,d^c=dh,e^c=ei,e^d=eh

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, 6530187);