#6483030

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1370 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)(11,15)(12,16)(13,14)(17,18,24,29)(19,30,34,44)(20,35,37,45)(21,27,39,42)(22,28,23,41)(25,32,43,33)(26,31,46,36)(38,47,40,48);(2,5)(6,8)(10,14)(11,12)(17,19,23,33,24,34,22,32)(18,25,28,30,29,43,41,44)(20,21,36,38,37,39,31,40)(26,42,35,47,46,27,45,48);(1,3,4,7)(2,6,5,8)(9,11,13,12)(10,15,14,16)(17,20,23,36,24,37,22,31)(18,26,28,35,29,46,41,45)(19,21,33,38,34,39,32,40)(25,42,30,47,43,27,44,48);(9,12,13,11)(10,16,14,15)(17,21)(18,27)(19,31)(20,33)(22,40)(23,38)(24,39)(25,35)(26,44)(28,48)(29,42)(30,46)(32,37)(34,36)(41,47)(43,45);(10,14)(15,16)(17,22,24,23)(18,28,29,41)(19,32,34,33)(20,36,37,31)(21,38,39,40)(25,30,43,44)(26,45,46,35)(27,47,42,48)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=f,d^2=h,e^2=i,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, 6390226);