#9863304

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

4026 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=gh,d^2=hi,e^2=ghi,f^2=h,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, 9770500);