#10195886

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

4210 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)(17,18,22,23)(19,25,26,20)(21,29,30,24)(27,28,32,31)(33,34)(35,43)(36,40)(37,46)(38,42)(39,45)(41,48)(44,47);(1,2,6,3)(4,7,12,9)(5,8,14,10)(11,15,13,16)(17,19,22,26)(18,20,23,25)(21,27,30,32)(24,28,29,31)(33,34,38,42)(35,39,45,43)(36,44,47,40)(37,41,48,46);(18,23)(19,26)(24,29)(27,32)(33,35,38,45)(34,39,42,43)(36,41,47,46)(37,44,48,40);(1,3,6,2)(4,7,12,9)(5,10,14,8)(11,15,13,16)(17,18)(19,20)(21,24)(22,23)(25,26)(27,28)(29,30)(31,32)(33,35)(34,39)(36,46)(37,44)(38,45)(40,48)(41,47)(42,43);(1,4,6,12)(2,7,3,9)(5,13,14,11)(8,16,10,15)(17,20)(18,19)(21,28)(22,25)(23,26)(24,27)(29,32)(30,31)(33,36)(34,40)(35,41)(37,39)(38,47)(42,44)(43,48)(45,46);(1,5,6,14)(2,8,3,10)(4,11,12,13)(7,15,9,16)(17,21,22,30)(18,24,23,29)(19,27,26,32)(20,28,25,31)(33,37)(34,41)(35,44)(36,43)(38,48)(39,47)(40,45)(42,46)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=hi,d^2=gh,e^2=gh,f^2=ghi,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, 10103082);