#6506952

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500535
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	#24270	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1660 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)(19,28)(20,25)(21,26)(22,37)(23,27)(24,32)(29,44)(30,45)(31,38)(33,48)(34,41)(35,43)(36,42)(39,46)(40,47);(2,5)(6,8)(9,10)(11,15)(12,14)(13,16)(19,29)(24,38)(28,44)(30,33)(31,32)(39,40)(45,48)(46,47);(9,11)(10,13)(12,16)(14,15)(17,19,21,31,23,29,22,32)(18,24,37,44,27,38,26,28)(20,33,36,46,35,30,34,47)(25,40,41,45,43,39,42,48);(1,3,4,7)(2,6,5,8)(11,16)(13,15)(17,20,23,35)(18,25,27,43)(19,30,29,33)(21,36,22,34)(24,39,38,40)(26,42,37,41)(28,45,44,48)(31,47,32,46);(10,14)(13,15)(17,21,23,22)(18,26,27,37)(19,31,29,32)(20,34,35,36)(24,28,38,44)(25,41,43,42)(30,46,33,47)(39,48,40,45)

Polycyclic presentation:

2^9|c^2=gi,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=ehi,d^c=dh,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6414148);