#7567912

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1300 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=fg,d^2=g,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dgi,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

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7475108);