#7505351

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500669
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 ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1791 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fg,c^2=f,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,h^b=hi,h^c=hi

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, 7412547);