#7505373

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500689
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×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1860 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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,c^b=ch,d^b=df,e^b=ei,h^b=hi,h^c=hi,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7412569);