#9910118

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

4198 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=gh,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9817314);