#6472902

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470901
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1602 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,24,30)(19,31,35,42)(20,36,37,45)(21,27,39,46)(22,29,23,28)(25,34,43,33)(26,38,44,32)(40,48,41,47);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19,22,33,24,35,23,34)(18,25,28,31,30,43,29,42)(20,21,38,40,37,39,32,41)(26,27,45,47,44,46,36,48);(1,3,4,7)(2,6,5,8)(11,15)(13,16)(17,20,22,38,24,37,23,32)(18,26,28,45,30,44,29,36)(19,21,33,40,35,39,34,41)(25,27,31,47,43,46,42,48);(11,15)(13,16)(17,21)(18,27)(19,32)(20,33)(22,40)(23,41)(24,39)(25,36)(26,31)(28,47)(29,48)(30,46)(34,37)(35,38)(42,44)(43,45);(10,14)(13,16)(20,37)(21,39)(26,44)(27,46)(32,38)(36,45)(40,41)(47,48)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=f,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dhi,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6380098);