#6465243

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438991
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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1369 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)(3,7)(4,6)(5,8)(17,18,23,28)(19,29,33,43)(20,34,36,45)(21,26,35,41)(22,39,30,27)(24,40,42,32)(25,31,44,37)(38,47,48,46);(1,3,5,4)(2,6,8,7)(9,10)(11,15)(12,13)(14,16)(17,19,30,40,23,33,22,32)(18,24,39,43,28,42,27,29)(20,35,31,38,36,21,37,48)(25,26,45,47,44,41,34,46);(1,3,5,4)(2,7,8,6)(10,13)(14,15)(17,20,22,37,23,36,30,31)(18,25,27,34,28,44,39,45)(19,21,32,38,33,35,40,48)(24,41,29,47,42,26,43,46);(9,11,12,16)(10,14,13,15)(17,21)(18,26)(19,31)(20,32)(22,38)(23,35)(24,34)(25,43)(27,46)(28,41)(29,44)(30,48)(33,37)(36,40)(39,47)(42,45);(1,4,5,3)(2,6,8,7)(20,36)(21,35)(25,44)(26,41)(31,37)(34,45)(38,48)(46,47)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=f,d^2=hi,e^2=fgi,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^c=ei,e^d=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, 6372439);