#8928740

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

4005 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=h,e^2=g,f^2=g,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, 8835936);