#10420121

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

4125 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)(3,9)(4,11)(5,8)(6,7)(10,16)(12,13)(14,15)(17,18,22,25)(19,23,28,21)(20,24,30,32)(26,31,27,29)(33,34,39,41)(35,36,43,37)(38,46,48,40)(42,45,47,44);(2,7)(4,9)(8,15)(12,16)(17,19,22,28)(18,23,25,21)(20,27,30,26)(24,29,32,31)(33,35)(34,36)(37,41)(38,47)(39,43)(40,45)(42,48)(44,46);(17,20)(18,24)(19,26)(21,29)(22,30)(23,31)(25,32)(27,28)(33,36,39,37)(34,35,41,43)(38,44,48,45)(40,42,46,47);(1,3,6,11)(2,4,7,9)(5,13,14,10)(8,16,15,12)(17,20)(18,24)(19,27)(21,31)(22,30)(23,29)(25,32)(26,28)(33,37,39,36)(34,35,41,43)(38,45,48,44)(40,42,46,47);(1,4)(2,3)(5,12)(6,9)(7,11)(8,10)(13,15)(14,16)(17,19)(18,21)(20,26)(22,28)(23,25)(24,29)(27,30)(31,32)(33,38)(34,40)(35,42)(36,44)(37,45)(39,48)(41,46)(43,47);(1,5,6,14)(2,8,7,15)(3,10,11,13)(4,12,9,16)(17,21)(18,19)(20,29)(22,23)(24,26)(25,28)(27,32)(30,31)(33,35)(34,37)(36,41)(38,42)(39,43)(40,45)(44,46)(47,48)

Polycyclic presentation:

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

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, 10327317);