#9711917

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

4112 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)(3,4)(5,13)(6,7)(8,15)(9,12)(10,14)(11,16)(17,18,23,25)(19,29,30,24)(20,21,27,26)(22,32,31,28)(34,40)(36,41)(42,47)(44,48);(1,3,6,9)(2,4,7,12)(5,10,15,16)(8,11,13,14)(17,19,23,30)(18,24,25,29)(20,28,27,32)(21,31,26,22)(33,34)(35,36)(37,41)(38,42)(39,40)(43,44)(45,48)(46,47);(2,7)(3,9)(8,13)(10,16)(18,25)(20,27)(24,29)(28,32)(33,35,39,37)(34,36,40,41)(38,43,46,45)(42,44,47,48);(1,2)(3,4)(5,8)(6,7)(9,12)(10,11)(13,15)(14,16)(17,20,23,27)(18,26,25,21)(19,28,30,32)(22,29,31,24)(33,36,39,41)(34,35,40,37)(38,44,46,48)(42,43,47,45);(1,4)(2,3)(5,14)(6,12)(7,9)(8,16)(10,13)(11,15)(17,21,23,26)(18,27,25,20)(19,22,30,31)(24,28,29,32)(33,37,39,35)(34,41,40,36)(38,43,46,45)(42,44,47,48);(1,5)(2,8)(3,10)(4,11)(6,15)(7,13)(9,16)(12,14)(17,22)(18,28)(19,21)(20,29)(23,31)(24,27)(25,32)(26,30)(33,38,39,46)(34,42,40,47)(35,43,37,45)(36,44,41,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=ghi,d^2=g,e^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=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, 9619113);