#6507376

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500450
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	#25109	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1661 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)(3,6)(4,5)(7,8)(17,18,23,27)(19,28,29,39)(20,25,32,40)(21,26,22,36)(24,34,37,31)(30,43,44,48)(33,41,35,42)(38,45,47,46);(2,5)(6,8)(9,10)(11,15)(12,14)(13,16)(19,29)(20,32)(24,37)(25,40)(28,39)(31,34)(33,35)(41,42);(9,11)(10,13)(12,16)(14,15)(17,19,21,34,23,29,22,31)(18,24,36,39,27,37,26,28)(20,30,35,45,32,44,33,46)(25,38,41,48,40,47,42,43);(1,3,4,7)(2,6,5,8)(11,16)(13,15)(17,20,23,32)(18,25,27,40)(19,30,29,44)(21,35,22,33)(24,38,37,47)(26,42,36,41)(28,43,39,48)(31,46,34,45);(10,14)(13,15)(17,21,23,22)(18,26,27,36)(19,31,29,34)(20,33,32,35)(24,39,37,28)(25,41,40,42)(30,45,44,46)(38,43,47,48)

Polycyclic presentation:

2^9|a^2=i,c^2=gi,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=ehi,d^c=dhi,e^c=ei,e^d=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, 6414572);