#6461258

order	512
nilpotency class	3
isoclinism	isoclinic class of #24390
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂³×D₄
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1960 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)(3,7)(4,5)(6,8)(9,10,12,14)(11,13,15,16)(17,18,23,37,22,27,21,26)(19,28,31,24,30,41,32,40)(20,33,38,43,35,25,36,48)(29,45,46,42,34,44,47,39);(2,5)(6,7)(9,11)(10,13)(12,15)(14,16)(17,19)(18,24)(20,34)(21,31)(22,30)(23,32)(25,42)(26,41)(27,40)(28,37)(29,35)(33,39)(36,47)(38,46)(43,44)(45,48);(1,3,4,8)(2,6,5,7)(10,14)(11,15)(17,20)(18,25)(19,29)(21,36)(22,35)(23,38)(24,39)(26,43)(27,33)(28,44)(30,34)(31,46)(32,47)(37,48)(40,42)(41,45);(11,15)(13,16)(19,30)(20,35)(24,40)(25,33)(28,41)(31,32)(36,38)(43,48);(17,21,22,23)(18,26,27,37)(19,31,30,32)(20,36,35,38)(24,41,40,28)(25,43,33,48)(29,46,34,47)(39,45,42,44)

Polycyclic presentation:

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

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, 6368454);