#8696275

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

4224 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,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, 8603471);