#6519382

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500535
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1433 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)(17,18,23,28)(19,25,31,42)(20,32,30,24)(21,26,37,45)(22,39,33,27)(29,41,35,44)(34,43,36,40)(38,48,47,46);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,30,23,31,33,20)(18,24,27,25,28,32,39,42)(21,35,38,34,37,29,47,36)(26,43,46,44,45,40,48,41);(1,3,4,7)(2,6,5,8)(10,13)(11,15)(17,20,33,31,23,30,22,19)(18,25,39,24,28,42,27,32)(21,36,47,29,37,34,38,35)(26,44,48,43,45,41,46,40);(9,11,12,15)(10,14,13,16)(17,21)(18,26)(19,29)(20,34)(22,38)(23,37)(24,40)(25,41)(27,46)(28,45)(30,36)(31,35)(32,43)(33,47)(39,48)(42,44);(10,13)(14,16)(21,37)(26,45)(29,35)(34,36)(38,47)(40,43)(41,44)(46,48)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6426578);