#8746603

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	#2414	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3875 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)(3,7)(4,9)(5,10)(6,11)(8,12)(13,15)(14,16)(17,18,22,21)(19,24,27,23)(20,28,29,25)(26,30,32,31)(33,34,39,37)(35,40,38,46)(36,43,45,41)(42,48,44,47);(1,3)(2,7)(4,13)(5,14)(6,12)(8,11)(9,15)(10,16)(17,19,22,27)(18,23,21,24)(20,26,29,32)(25,30,28,31)(33,35,39,38)(34,40,37,46)(36,42,45,44)(41,47,43,48);(3,12)(7,8)(13,16)(14,15)(17,19,22,27)(18,23,21,24)(20,26,29,32)(25,30,28,31)(33,36)(34,41)(35,42)(37,43)(38,44)(39,45)(40,47)(46,48);(1,3)(2,8)(4,13)(5,15)(6,12)(7,11)(9,14)(10,16)(17,19)(18,24)(20,26)(21,23)(22,27)(25,31)(28,30)(29,32)(33,37,39,34)(35,40,38,46)(36,43,45,41)(42,47,44,48);(1,4)(2,9)(3,13)(5,11)(6,10)(7,15)(8,14)(12,16)(17,20,22,29)(18,25,21,28)(19,26,27,32)(23,30,24,31)(33,38,39,35)(34,40,37,46)(36,44,45,42)(41,47,43,48);(1,5,6,9)(2,10,11,4)(3,14,12,15)(7,16,8,13)(17,21,22,18)(19,24,27,23)(20,25,29,28)(26,30,32,31)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=g,d^2=gh,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8653799);