#8434132

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

4281 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,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, 8341328);