#8750422

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	C₂×#127	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3879 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)(3,7)(4,12)(5,9)(6,10)(8,13)(11,14)(15,16)(17,18,23,27)(19,26,21,24)(20,25,31,22)(28,32,30,29)(33,34,38,39)(35,36,41,44)(37,40,45,46)(42,43,47,48);(1,3)(2,7)(4,9)(5,12)(6,11)(8,16)(10,14)(13,15)(17,19,23,21)(18,24,27,26)(20,30,31,28)(22,29,25,32)(34,39)(35,41)(40,46)(42,47);(3,11)(4,13)(7,14)(8,12)(17,19,23,21)(18,24,27,26)(20,30,31,28)(22,29,25,32)(33,35)(34,36)(37,42)(38,41)(39,44)(40,43)(45,47)(46,48);(1,4,6,13)(2,8,10,12)(3,9,11,15)(5,7,16,14)(17,20,23,31)(18,25,27,22)(19,28,21,30)(24,29,26,32)(33,36)(34,35)(37,43)(38,44)(39,41)(40,42)(45,48)(46,47);(1,2)(3,7)(4,8)(5,15)(6,10)(9,16)(11,14)(12,13)(17,21,23,19)(18,26,27,24)(20,30,31,28)(22,29,25,32)(33,37,38,45)(34,40,39,46)(35,42,41,47)(36,43,44,48);(1,5,6,16)(2,9,10,15)(3,12,11,8)(4,14,13,7)(17,22,23,25)(18,20,27,31)(19,29,21,32)(24,30,26,28)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=gi,e^2=g,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, 8657618);