#10407857

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

4342 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:

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

Polycyclic presentation:

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

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, 10315053);