#10424608

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

4132 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gi,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=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, 10331804);