#6465339

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

1365 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)(19,29,34,43)(20,35,36,40)(21,26,31,45)(22,39,30,27)(24,37,41,33)(25,42,44,32)(38,48,47,46);(1,3,5,4)(2,6,8,7)(9,10)(11,15)(12,13)(14,16)(17,19,30,37,23,34,22,33)(18,24,39,43,28,41,27,29)(20,31,42,38,36,21,32,47)(25,26,40,48,44,45,35,46);(1,3,5,4)(2,7,8,6)(10,13)(14,15)(17,20,22,32,23,36,30,42)(18,25,27,35,28,44,39,40)(19,31,33,47,34,21,37,38)(24,26,29,46,41,45,43,48);(9,11,12,16)(10,14,13,15)(17,21,23,31)(18,26,28,45)(19,32,34,42)(20,33,36,37)(22,38,30,47)(24,40,41,35)(25,43,44,29)(27,46,39,48);(1,4,5,3)(2,6,8,7)(20,36)(21,31)(25,44)(26,45)(32,42)(35,40)(38,47)(46,48)

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,d^c=di,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, 6372535);