#9863289

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

3955 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)(18,22)(20,28)(23,25)(26,32)(33,34,39,36)(35,37,40,42)(38,41,47,44)(43,45,48,46);(1,3,7,11)(2,8,9,16)(4,12,13,6)(5,10,14,15)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(35,40)(37,42)(43,48)(45,46);(2,9)(3,11)(4,13)(10,15)(17,18,21,22)(19,23,27,25)(20,24,28,29)(26,30,32,31)(33,35)(34,37)(36,42)(38,43)(39,40)(41,45)(44,46)(47,48);(1,4)(2,5)(3,12)(6,11)(7,13)(8,10)(9,14)(15,16)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(33,36,39,34)(35,42,40,37)(38,44,47,41)(43,46,48,45);(1,5)(2,4)(3,10)(6,16)(7,14)(8,12)(9,13)(11,15)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31)(33,37,39,42)(34,40,36,35)(38,46,47,45)(41,43,44,48);(1,6,7,12)(2,10,9,15)(3,4,11,13)(5,8,14,16)(17,20)(18,24)(19,26)(21,28)(22,29)(23,30)(25,31)(27,32)(33,38)(34,41)(35,43)(36,44)(37,45)(39,47)(40,48)(42,46)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=g,d^2=hi,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, 9770485);