#7509450

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500669
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 ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1273 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)(3,4)(5,7)(6,8)(9,10)(11,14)(12,16)(13,15)(17,18)(19,23)(20,31)(21,24)(22,26)(25,32)(27,40)(28,42)(29,37)(30,35)(33,45)(34,38)(36,43)(39,46)(41,47)(44,48);(1,3)(2,6)(4,7)(5,8)(9,11,13,12)(10,14,15,16)(17,19,21,29)(18,23,24,37)(20,28,32,43)(22,27,34,30)(25,36,31,42)(26,40,38,35)(33,41,46,44)(39,48,45,47);(9,12,13,11)(10,14,15,16)(18,24)(19,27)(22,34)(23,35)(25,31)(28,41)(29,30)(33,46)(36,47)(37,40)(42,48)(43,44);(2,7)(3,8)(10,15)(14,16)(17,20,21,32)(18,25,24,31)(19,28,29,43)(22,33,34,46)(23,36,37,42)(26,39,38,45)(27,41,30,44)(35,47,40,48);(1,4,5,6)(2,3,7,8)(18,24)(20,32)(23,37)(26,38)(28,43)(33,46)(35,40)(41,44)

Polycyclic presentation:

2^9|b^2=g,c^2=gi,d^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,h^b=hi,h^c=hi,e^d=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, 7416646);