#8746941

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

4007 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)(4,9,12,7)(5,8,14,10)(11,16,13,15)(17,18,22,21)(19,23,26,24)(20,25,28,29)(27,30,32,31)(33,34,39,38)(35,42,37,40)(36,41,44,46)(43,48,45,47);(2,3)(7,9)(8,10)(15,16)(17,19)(18,23)(20,27)(21,24)(22,26)(25,30)(28,32)(29,31)(33,35,39,37)(34,40,38,42)(36,43,44,45)(41,47,46,48);(2,3)(7,9)(8,10)(15,16)(19,26)(23,24)(27,32)(30,31)(36,44)(41,46)(43,45)(47,48);(1,3,6,2)(4,9,12,7)(5,8,14,10)(11,15,13,16)(17,19)(18,24)(20,27)(21,23)(22,26)(25,31)(28,32)(29,30)(33,36)(34,41)(35,43)(37,45)(38,46)(39,44)(40,47)(42,48);(1,4,6,12)(2,7,3,9)(5,13,14,11)(8,16,10,15)(17,20)(18,25)(19,27)(21,29)(22,28)(23,30)(24,31)(26,32)(33,37,39,35)(34,40,38,42)(36,45,44,43)(41,47,46,48);(1,5,6,14)(2,8,3,10)(4,11,12,13)(7,15,9,16)(17,21,22,18)(19,24,26,23)(20,25,28,29)(27,30,32,31)(33,38,39,34)(35,42,37,40)(36,41,44,46)(43,47,45,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,d^2=h,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8654137);