#10336064

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

4288 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=ghi,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi

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, 10243260);