#8505074

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

4110 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)(3,8)(4,9)(5,10)(6,11)(7,12)(13,16)(14,15)(17,18,23,20)(19,24,29,27)(21,30,32,25)(22,26,28,31)(33,34,39,43)(35,44,38,40)(36,41,42,37)(45,48,47,46);(3,12)(4,14)(7,8)(9,15)(17,19)(18,24)(20,27)(21,22)(23,29)(25,31)(26,30)(28,32)(33,35,39,38)(34,40,43,44)(36,45,42,47)(37,48,41,46);(1,3)(2,7)(4,13)(5,9)(6,12)(8,11)(10,14)(15,16)(17,18,23,20)(19,27,29,24)(21,25,32,30)(22,26,28,31)(33,36,39,42)(34,41,43,37)(35,45,38,47)(40,46,44,48);(1,3)(2,8)(4,13)(5,15)(6,12)(7,11)(9,16)(10,14)(17,20,23,18)(19,27,29,24)(21,30,32,25)(22,26,28,31)(33,34)(35,40)(36,41)(37,42)(38,44)(39,43)(45,46)(47,48);(1,4)(2,9)(3,13)(5,7)(6,14)(8,16)(10,12)(11,15)(17,21)(18,25)(19,28)(20,30)(22,29)(23,32)(24,26)(27,31)(33,37,39,41)(34,42,43,36)(35,46,38,48)(40,45,44,47);(1,5,6,16)(2,10,11,13)(3,9,12,15)(4,7,14,8)(17,22,23,28)(18,26,20,31)(19,21,29,32)(24,30,27,25)(33,38,39,35)(34,40,43,44)(36,47,42,45)(37,48,41,46)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8412270);