#6464247

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	#23386	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1925 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)(17,18,22,28,23,26,21,27)(19,29,31,38,32,42,36,24)(20,25,35,40,33,41,34,39)(30,43,45,47,44,48,46,37);(9,11)(10,13)(12,15)(14,16)(17,19)(18,24)(20,30)(21,31)(22,36)(23,32)(25,37)(26,38)(27,29)(28,42)(33,44)(34,45)(35,46)(39,43)(40,48)(41,47);(2,5)(6,8)(10,14)(11,15)(17,20)(18,25)(19,30)(21,34)(22,35)(23,33)(24,37)(26,41)(27,39)(28,40)(29,43)(31,45)(32,44)(36,46)(38,47)(42,48);(1,3,4,7)(2,6,5,8)(11,15)(13,16)(18,26)(20,33)(24,38)(27,28)(29,42)(30,44)(34,35)(45,46);(17,21,23,22)(18,27,26,28)(19,31,32,36)(20,34,33,35)(24,29,38,42)(25,39,41,40)(30,45,44,46)(37,43,47,48)

Polycyclic presentation:

2^9|a^2=f,d^2=gh,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,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, 6371443);