#8730836

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

4150 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=gh,d^2=g,e^2=g,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, 8638032);