#8902513

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

4214 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,d^2=hi,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8809709);