#10407860

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

4242 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,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, 10315056);