#8463664

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

3985 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,d^2=g,e^2=ghi,f^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, 8370860);