#8425463

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

4194 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=hi,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,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, 8332659);