#6468846

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437758
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	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1819 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=i,e^2=gh,f^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,e^c=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6376042);