#9851110

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

3960 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)(17,18,22,19)(20,23,28,25)(21,26,30,24)(27,32,29,31)(33,34,38,42)(35,39,43,47)(36,40,46,45)(37,41,48,44);(1,2,6,3)(4,9,12,7)(5,8,14,10)(11,16,13,15)(17,18,22,19)(20,23,28,25)(21,24,30,26)(27,31,29,32)(35,43)(36,46)(39,47)(40,45);(18,19)(23,25)(24,26)(31,32)(33,35,38,43)(34,39,42,47)(36,44,46,41)(37,45,48,40);(1,3,6,2)(4,9,12,7)(5,10,14,8)(11,16,13,15)(17,19,22,18)(20,25,28,23)(21,26,30,24)(27,32,29,31);(1,4,6,12)(2,7,3,9)(5,13,14,11)(8,16,10,15)(17,20,22,28)(18,23,19,25)(21,29,30,27)(24,32,26,31)(33,36,38,46)(34,40,42,45)(35,44,43,41)(37,47,48,39);(1,5,6,14)(2,8,3,10)(4,11,12,13)(7,15,9,16)(17,21)(18,24)(19,26)(20,27)(22,30)(23,31)(25,32)(28,29)(33,37)(34,41)(35,45)(36,39)(38,48)(40,43)(42,44)(46,47)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=hi,d^2=gh,e^2=gi,f^2=g,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, 9758306);