#8496849

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

4169 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)(3,9)(4,12)(5,14)(6,11)(7,8)(10,15)(13,16)(17,18,23,22)(19,24,29,28)(20,30,21,25)(26,32,27,31)(33,34,39,37)(35,42,38,40)(36,45,46,41)(43,47,44,48);(2,8)(5,15)(9,12)(11,13)(17,19,23,29)(18,24,22,28)(20,27,21,26)(25,32,30,31)(33,35,39,38)(34,40,37,42)(36,43,46,44)(41,47,45,48);(1,3,7,4)(2,9,8,12)(5,11,15,13)(6,10,16,14)(17,20,23,21)(18,25,22,30)(19,26,29,27)(24,31,28,32)(33,36)(34,41)(35,43)(37,45)(38,44)(39,46)(40,47)(42,48);(1,4,7,3)(2,9,8,12)(5,13,15,11)(6,10,16,14)(17,21,23,20)(18,25,22,30)(19,27,29,26)(24,31,28,32)(33,36)(34,41)(35,44)(37,45)(38,43)(39,46)(40,48)(42,47);(1,5,7,15)(2,10,8,14)(3,11,4,13)(6,9,16,12)(17,19)(18,24)(20,26)(21,27)(22,28)(23,29)(25,31)(30,32)(33,37,39,34)(35,40,38,42)(36,45,46,41)(43,47,44,48);(1,6,7,16)(2,11,8,13)(3,10,4,14)(5,9,15,12)(17,22,23,18)(19,28,29,24)(20,30,21,25)(26,32,27,31)(33,38,39,35)(34,40,37,42)(36,44,46,43)(41,47,45,48)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gh,d^2=gh,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=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, 8404045);