#7575300

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	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1274 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=f,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,e^c=ei,e^d=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, 7482496);