#8757067

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

4133 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)(17,18,24,29)(19,27,21,25)(20,26,22,31)(23,28,32,30)(34,36)(35,41)(37,45)(46,48);(1,2)(3,10)(4,12)(5,8)(6,9)(7,13)(11,14)(15,16)(17,19,24,21)(18,25,29,27)(20,30,22,28)(23,26,32,31)(33,34,38,36)(35,39,41,42)(37,44,45,40)(43,48,47,46);(1,2,5,8)(3,6,9,10)(4,7,13,12)(11,14,15,16)(17,20,24,22)(18,26,29,31)(19,30,21,28)(23,25,32,27)(33,34,38,36)(35,42,41,39)(37,40,45,44)(43,48,47,46);(1,2)(3,6)(4,7)(5,8)(9,10)(11,14)(12,13)(15,16)(17,21,24,19)(18,27,29,25)(20,30,22,28)(23,26,32,31)(33,35,38,41)(34,39,36,42)(37,43,45,47)(40,46,44,48);(1,3,5,9)(2,6,8,10)(4,11,13,15)(7,14,12,16)(17,22,24,20)(18,26,29,31)(19,30,21,28)(23,27,32,25)(33,36,38,34)(35,42,41,39)(37,44,45,40)(43,48,47,46);(1,4)(2,7)(3,11)(5,13)(6,14)(8,12)(9,15)(10,16)(17,23,24,32)(18,28,29,30)(19,31,21,26)(20,25,22,27)(33,37)(34,40)(35,43)(36,44)(38,45)(39,46)(41,47)(42,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=gi,d^2=g,e^2=gi,f^2=gh,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, 8664263);