#8696272

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

4202 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=gi,d^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=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, 8603468);