#9910237

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

4210 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)(3,13,14,8)(4,5,11,10)(6,16,15,12)(17,18,23,21)(19,25,20,24)(22,26,30,29)(27,32,28,31)(33,34)(35,38)(36,43)(37,40)(39,44)(41,47)(42,46)(45,48);(1,3)(2,8)(4,15)(5,16)(6,11)(7,14)(9,13)(10,12)(17,19,23,20)(18,24,21,25)(22,27,30,28)(26,31,29,32)(33,35,37,42)(34,38,40,46)(36,41,44,48)(39,45,43,47);(2,9)(3,14)(4,11)(12,16)(17,19,23,20)(18,25,21,24)(22,27,30,28)(26,32,29,31)(33,34,37,40)(35,38,42,46)(36,39,44,43)(41,45,48,47);(1,4,7,11)(2,10,9,5)(3,15,14,6)(8,12,13,16)(17,20,23,19)(18,24,21,25)(22,28,30,27)(26,31,29,32)(33,34)(35,38)(36,39)(37,40)(41,45)(42,46)(43,44)(47,48);(1,5,7,10)(2,11,9,4)(3,12,14,16)(6,8,15,13)(17,21,23,18)(19,24,20,25)(22,26,30,29)(27,32,28,31)(33,35)(34,38)(36,41)(37,42)(39,45)(40,46)(43,47)(44,48);(1,6,7,15)(2,12,9,16)(3,11,14,4)(5,8,10,13)(17,22,23,30)(18,26,21,29)(19,27,20,28)(24,31,25,32)(33,36)(34,39)(35,41)(37,44)(38,45)(40,43)(42,48)(46,47)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=gh,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9817433);