#8459040

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	#45522	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4091 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)(3,13)(4,10)(5,7)(6,16)(8,12)(9,15)(11,14)(20,28)(24,31)(27,32)(29,30)(33,34,39,42)(35,43,46,38)(36,37,41,40)(44,47,48,45);(1,3,8,15)(2,9,12,13)(4,14,5,6)(7,11,10,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32)(33,35,39,46)(34,38,42,43)(36,44,41,48)(37,45,40,47);(1,4,8,5)(2,10,12,7)(3,14,15,6)(9,16,13,11)(17,19)(18,21)(20,27)(22,25)(23,26)(24,29)(28,32)(30,31)(33,36,39,41)(34,40,42,37)(35,44,46,48)(38,47,43,45);(1,5,8,4)(2,10,12,7)(3,6,15,14)(9,16,13,11)(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,32)(29,31)(33,34)(35,43)(36,40)(37,41)(38,46)(39,42)(44,45)(47,48);(1,6)(2,11)(3,4)(5,15)(7,13)(8,14)(9,10)(12,16)(17,20,22,28)(18,24,26,31)(19,27,25,32)(21,29,23,30)(33,37,39,40)(34,41,42,36)(35,45,46,47)(38,48,43,44);(1,7,8,10)(2,5,12,4)(3,11,15,16)(6,13,14,9)(17,21,22,23)(18,25,26,19)(20,29,28,30)(24,32,31,27)(33,38,39,43)(34,35,42,46)(36,47,41,45)(37,48,40,44)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,d^2=gh,e^2=hi,f^2=gi,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, 8366236);