#8630890

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

4158 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=i,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8538086);