#8691861

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

4158 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=h,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8599057);