#8818273

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

4155 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)(3,11)(4,9)(5,10)(6,7)(8,12)(13,14)(15,16)(17,18)(19,23)(20,28)(21,22)(24,29)(25,26)(27,31)(30,32);(2,7)(5,15)(8,11)(9,13)(17,18)(19,25)(20,28)(21,22)(23,26)(24,29)(27,30)(31,32)(33,34)(35,38)(36,44)(37,40)(39,45)(41,43)(42,47)(46,48);(3,12)(4,14)(8,11)(9,13)(18,22)(19,26)(24,28)(27,32)(33,34,37,40)(35,41,43,38)(36,39,45,44)(42,47,48,46);(1,3)(2,8)(4,10)(5,13)(6,12)(7,11)(9,15)(14,16)(17,18)(19,23)(20,24)(21,22)(25,26)(27,30)(28,29)(31,32)(33,35,37,43)(34,38,40,41)(36,42,45,48)(39,46,44,47);(1,4)(2,9)(3,10)(5,11)(6,14)(7,13)(8,15)(12,16)(17,19,21,26)(18,23,22,25)(20,27,29,32)(24,30,28,31)(33,34)(35,38)(36,39)(37,40)(41,43)(42,46)(44,45)(47,48);(1,5,6,15)(2,10,7,16)(3,13,12,9)(4,11,14,8)(17,20,21,29)(18,24,22,28)(19,27,26,32)(23,30,25,31)(33,36,37,45)(34,39,40,44)(35,42,43,48)(38,46,41,47)

Polycyclic presentation:

2^9|c^2=hi,d^2=hi,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8725469);