#7575155

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,c^2=fi,d^2=i,e^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,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, 7482351);