#10534339

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

3987 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=gi,d^2=g,e^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10441535);