#8446075

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
prime radical	#38483	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4048 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=g,e^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,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, 8353271);