#9835833

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

4018 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)(3,7,10,4)(5,13,15,9)(11,12,16,14)(17,18)(19,26)(20,28)(21,31)(22,24)(23,29)(25,30)(27,32)(33,34,38,42)(35,43,45,39)(36,46,47,40)(37,41,44,48);(1,3)(2,7)(4,8)(5,11)(6,10)(9,14)(12,13)(15,16)(17,19)(18,23)(20,27)(21,28)(22,29)(24,26)(25,32)(30,31)(33,35,38,45)(34,39,42,43)(36,37,47,44)(40,41,46,48);(2,8)(3,10)(9,13)(11,16)(18,24)(20,30)(23,26)(27,31)(33,36,38,47)(34,40,42,46)(35,44,45,37)(39,48,43,41);(1,3)(2,7)(4,8)(5,11)(6,10)(9,14)(12,13)(15,16)(17,20,22,30)(18,25,24,28)(19,27,29,31)(21,23,32,26)(33,36)(34,40)(35,37)(38,47)(39,41)(42,46)(43,48)(44,45);(1,4,6,7)(2,3,8,10)(5,14,15,12)(9,16,13,11)(17,19)(18,26)(20,27)(21,25)(22,29)(23,24)(28,32)(30,31)(33,37)(34,41)(35,36)(38,44)(39,40)(42,48)(43,46)(45,47);(1,5,6,15)(2,9,8,13)(3,11,10,16)(4,12,7,14)(17,21)(18,27)(19,28)(20,23)(22,32)(24,31)(25,29)(26,30)(33,34)(35,39)(36,40)(37,41)(38,42)(43,45)(44,48)(46,47)

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=ghi,d^2=hi,e^2=gh,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, 9743029);