#8756990

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	#49851	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3981 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=i,d^2=gh,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8664186);