#7620575

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
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

1255 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fi,d^2=fgh,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7527771);