#9175029

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4091 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=hi,c^2=gh,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9082225);