#8179670

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

4154 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)(3,11)(4,6)(5,8)(7,9)(10,15)(12,16)(13,14)(17,18)(19,23)(20,25)(21,29)(22,24)(26,30)(27,28)(31,32)(33,34,38,37)(35,41,43,39)(36,44,45,40)(42,46,48,47);(1,3,7,5)(2,8,9,11)(4,12,15,14)(6,13,10,16)(17,19,22,25)(18,23,24,20)(21,27,30,32)(26,31,29,28)(33,35,38,43)(34,39,37,41)(36,42,45,48)(40,46,44,47);(2,9)(6,10)(8,11)(13,16)(18,24)(19,25)(26,29)(27,32)(33,35)(34,39)(36,42)(37,41)(38,43)(40,46)(44,47)(45,48);(1,4)(2,10)(3,12)(5,14)(6,9)(7,15)(8,16)(11,13)(17,20,22,23)(18,25,24,19)(21,28,30,31)(26,32,29,27)(33,35)(34,39)(36,42)(37,41)(38,43)(40,46)(44,47)(45,48);(1,5,7,3)(2,11,9,8)(4,14,15,12)(6,16,10,13)(17,21,22,30)(18,26,24,29)(19,27,25,32)(20,28,23,31)(33,36,38,45)(34,40,37,44)(35,42,43,48)(39,46,41,47);(1,6,7,10)(2,4,9,15)(3,13,5,16)(8,12,11,14)(17,18)(19,23)(20,25)(21,26)(22,24)(27,31)(28,32)(29,30)(33,37,38,34)(35,41,43,39)(36,44,45,40)(42,47,48,46)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8086866);