#9160800

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
prime radical	#40392	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4099 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9067996);