#8696716

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

4100 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)(3,10)(4,7)(5,8)(6,9)(11,16)(12,14)(13,15)(17,18,22,21)(19,24,27,23)(20,28,29,25)(26,30,32,31)(33,34,39,43)(35,40,44,46)(36,41,45,48)(37,38,42,47);(1,3,7,5)(2,8,4,10)(6,11,14,13)(9,15,12,16)(17,18,22,21)(19,24,27,23)(20,25,29,28)(26,31,32,30)(35,44)(36,45)(40,46)(41,48);(17,19)(18,23)(20,26)(21,24)(22,27)(25,30)(28,31)(29,32)(33,35,39,44)(34,40,43,46)(36,38,45,47)(37,41,42,48);(1,4,7,2)(3,8,5,10)(6,9,14,12)(11,16,13,15)(17,19)(18,24)(20,26)(21,23)(22,27)(25,31)(28,30)(29,32)(33,36)(34,41)(35,38)(37,46)(39,45)(40,42)(43,48)(44,47);(1,5,7,3)(2,8,4,10)(6,11,14,13)(9,16,12,15)(17,20)(18,25)(19,26)(21,28)(22,29)(23,30)(24,31)(27,32)(33,37)(34,38)(35,41)(36,46)(39,42)(40,45)(43,47)(44,48);(1,6,7,14)(2,9,4,12)(3,11,5,13)(8,15,10,16)(17,21,22,18)(19,24,27,23)(20,25,29,28)(26,30,32,31)(33,38)(34,42)(35,45)(36,44)(37,43)(39,47)(40,48)(41,46)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=hi,d^2=g,e^2=g,f^2=gh,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, 8603912);