#10534382

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

4065 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=hi,d^2=g,e^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10441578);