#10143455

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

4043 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)(4,7,11,9)(5,10,13,8)(12,15,16,14)(17,18)(19,24)(20,26)(21,27)(22,32)(23,28)(25,31)(29,30)(34,40)(36,47)(41,43)(44,45);(4,11)(7,9)(12,16)(14,15)(17,19,23,29)(18,24,28,30)(20,27,31,22)(21,25,32,26)(33,34,39,40)(35,41,37,43)(36,42,47,46)(38,44,48,45);(2,3)(7,9)(8,10)(14,15)(17,20,23,31)(18,25,28,26)(19,22,29,27)(21,30,32,24)(33,35,39,37)(34,41,40,43)(36,45,47,44)(38,46,48,42);(1,3,6,2)(4,9,11,7)(5,10,13,8)(12,15,16,14)(17,20)(18,26)(19,27)(21,24)(22,29)(23,31)(25,28)(30,32)(33,36)(34,42)(35,45)(37,44)(38,41)(39,47)(40,46)(43,48);(1,4,6,11)(2,7,3,9)(5,12,13,16)(8,14,10,15)(17,21,23,32)(18,27,28,22)(19,26,29,25)(20,30,31,24)(33,37,39,35)(34,43,40,41)(36,45,47,44)(38,46,48,42);(1,5,6,13)(2,8,3,10)(4,12,11,16)(7,14,9,15)(17,22,23,27)(18,21,28,32)(19,20,29,31)(24,25,30,26)(33,38)(34,44)(35,46)(36,41)(37,42)(39,48)(40,45)(43,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=gi,d^2=hi,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10050651);