#10128131

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

4064 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)(19,23)(20,25)(21,28)(22,24)(26,32)(27,30)(29,31)(33,34,39,44)(35,43,38,40)(36,37,42,41)(45,46,47,48);(4,11)(7,9)(12,16)(14,15)(17,19,22,27)(18,23,24,30)(20,26,31,28)(21,25,32,29)(33,35,39,38)(34,40,44,43)(36,45,42,47)(37,48,41,46);(2,3)(7,9)(8,10)(14,15)(18,24)(19,27)(25,29)(26,28)(33,36,39,42)(34,41,44,37)(35,45,38,47)(40,46,43,48);(1,3,6,2)(4,9,11,7)(5,10,13,8)(12,15,16,14)(17,18)(19,23)(20,29)(21,26)(22,24)(25,31)(27,30)(28,32)(33,36)(34,41)(35,45)(37,44)(38,47)(39,42)(40,46)(43,48);(1,4)(2,7)(3,9)(5,12)(6,11)(8,14)(10,15)(13,16)(17,20)(18,25)(19,28)(21,23)(22,31)(24,29)(26,27)(30,32)(33,37,39,41)(34,42,44,36)(35,46,38,48)(40,45,43,47);(1,5)(2,8)(3,10)(4,12)(6,13)(7,14)(9,15)(11,16)(17,21)(18,26)(19,25)(20,30)(22,32)(23,31)(24,28)(27,29)(33,38,39,35)(34,43,44,40)(36,47,42,45)(37,46,41,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=g,d^2=ghi,e^2=g,f^2=g,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, 10035327);