#9863296

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

3941 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=g,d^2=ghi,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,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, 9770492);