#8931482

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

4117 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:

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

Polycyclic presentation:

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

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, 8838678);