#7542577

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514020
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1341 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)(3,7)(4,8)(5,6)(9,10)(11,14)(12,16)(13,15)(17,18,23,29)(19,30,34,25)(20,27,21,26)(22,39,24,28)(31,40,36,43)(32,41,48,47)(33,42,35,46)(37,44,38,45);(2,6)(7,8)(9,11,13,16)(10,14,15,12)(17,19)(18,25)(20,36)(21,31)(22,35)(23,34)(24,33)(26,43)(27,40)(28,46)(29,30)(32,38)(37,48)(39,42)(41,44)(45,47);(1,3,5,4)(2,7,6,8)(9,12,13,14)(10,11,15,16)(17,20,23,21)(18,26,29,27)(19,31,34,36)(22,38,24,37)(25,40,30,43)(28,45,39,44)(32,35,48,33)(41,42,47,46);(10,15)(11,16)(17,21,23,20)(18,27,29,26)(19,32,34,48)(22,38,24,37)(25,41,30,47)(28,45,39,44)(31,35,36,33)(40,42,43,46);(1,4,5,3)(2,7,6,8)(17,22,23,24)(18,28,29,39)(19,33,34,35)(20,37,21,38)(25,42,30,46)(26,44,27,45)(31,48,36,32)(40,41,43,47)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=fgi,d^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,e^b=ei,h^b=hi,e^c=ei,h^d=hi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7449773);