#8436218

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	#2543	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3994 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)(3,6)(4,9)(5,8)(7,12)(10,13)(11,15)(14,16)(17,18,23,22)(19,26,29,24)(20,25,21,30)(27,32,28,31)(33,34,38,41)(35,39,43,37)(36,44,46,40)(42,45,48,47);(2,3)(7,9)(8,10)(14,15)(17,19)(18,24)(20,27)(21,28)(22,26)(23,29)(25,31)(30,32)(33,35,38,43)(34,37,41,39)(36,42,46,48)(40,45,44,47);(1,2,6,3)(4,7,12,9)(5,8,13,10)(11,14,16,15)(17,20,23,21)(18,25,22,30)(19,27,29,28)(24,31,26,32)(33,35)(34,39)(36,42)(37,41)(38,43)(40,47)(44,45)(46,48);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,21,23,20)(18,25,22,30)(19,28,29,27)(24,31,26,32)(33,36,38,46)(34,40,41,44)(35,42,43,48)(37,45,39,47);(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,15)(13,16)(17,19)(18,24)(20,27)(21,28)(22,26)(23,29)(25,31)(30,32);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,22,23,18)(19,26,29,24)(20,30,21,25)(27,32,28,31)(33,37,38,39)(34,35,41,43)(36,45,46,47)(40,42,44,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=ghi,d^2=g,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, 8343414);