#10143780

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

4234 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=hi,d^2=ghi,e^2=gh,f^2=gh,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, 10050976);