#8455931

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	#46256	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4092 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=hi,e^2=gi,f^2=g,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, 8363127);