#8751468

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	#2540	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3883 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=ghi,d^2=gi,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8658664);