#9851071

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

4091 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=ghi,d^2=gh,e^2=gi,f^2=g,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, 9758267);