#8425446

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

4240 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)(3,6)(4,9)(5,8)(7,12)(10,13)(11,15)(14,16)(17,18,22,23)(19,20,27,24)(21,25,30,31)(26,28,32,29)(33,34,39,45)(35,40,37,42)(36,44,47,41)(38,46,48,43);(2,3)(7,9)(8,10)(14,15)(18,23)(20,24)(25,31)(28,29)(33,35)(34,40)(36,46)(37,39)(38,41)(42,45)(43,47)(44,48);(1,2,6,3)(4,7,12,9)(5,8,13,10)(11,14,16,15)(17,19)(18,24)(20,23)(21,26)(22,27)(25,29)(28,31)(30,32)(33,36)(34,41)(35,43)(37,46)(38,42)(39,47)(40,48)(44,45);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,20,22,24)(18,19,23,27)(21,29,30,28)(25,32,31,26)(33,37,39,35)(34,42,45,40)(36,46,47,43)(38,44,48,41);(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,15)(13,16)(17,19)(18,24)(20,23)(21,26)(22,27)(25,29)(28,31)(30,32)(33,38)(34,43)(35,41)(36,42)(37,44)(39,48)(40,47)(45,46);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,21,22,30)(18,25,23,31)(19,26,27,32)(20,28,24,29)(33,36)(34,44)(35,46)(37,43)(38,42)(39,47)(40,48)(41,45)

Polycyclic presentation:

2^9|a^2=gi,c^2=ghi,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=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, 8332642);