#9863141

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

4079 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=h,d^2=gi,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,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, 9770337);