#8436214

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	#2430	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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=hi,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, 8343410);