#8533518

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

4256 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)(3,9)(4,6)(5,8)(7,11)(10,16)(12,14)(13,15)(17,18)(19,20)(21,25)(22,23)(24,26)(27,28)(29,32)(30,31)(33,34,40,44)(35,45,48,41)(36,42,37,38)(39,43,47,46);(1,3,6,11)(2,7,4,9)(5,10,14,13)(8,15,12,16)(17,19)(18,20)(21,27)(22,26)(23,24)(25,28)(29,31)(30,32)(33,35)(34,41)(36,43)(37,46)(38,47)(39,42)(40,48)(44,45);(2,4)(7,9)(8,12)(15,16)(18,23)(19,26)(25,31)(27,32)(33,36,40,37)(34,42,44,38)(35,43,48,46)(39,45,47,41);(1,3)(2,7)(4,9)(5,13)(6,11)(8,16)(10,14)(12,15)(17,20,22,24)(18,19,23,26)(21,29,30,28)(25,32,31,27)(33,37,40,36)(34,42,44,38)(35,46,48,43)(39,45,47,41);(1,4,6,2)(3,9,11,7)(5,12,14,8)(10,16,13,15)(17,19)(18,24)(20,23)(21,27)(22,26)(25,29)(28,31)(30,32)(33,38,40,42)(34,36,44,37)(35,47,48,39)(41,43,45,46);(1,5)(2,8)(3,10)(4,12)(6,14)(7,15)(9,16)(11,13)(17,21,22,30)(18,25,23,31)(19,27,26,32)(20,28,24,29)(33,39,40,47)(34,43,44,46)(35,42,48,38)(36,45,37,41)

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=ghi,d^2=gh,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=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, 8440714);