#8746000

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	#2456	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3809 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:

(1,2,7,11)(3,12,14,8)(4,10,15,9)(5,16,13,6)(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)(2,8)(4,5)(6,10)(7,14)(9,16)(11,12)(13,15)(17,18,22,26)(19,23,27,30)(20,24,29,32)(21,25,28,31)(34,38)(39,41)(40,45)(46,47);(1,4,7,15)(2,9,11,10)(3,5,14,13)(6,8,16,12)(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)(2,10)(3,13)(5,14)(6,12)(7,15)(8,16)(9,11)(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);(1,5,7,13)(2,6,11,16)(3,4,14,15)(8,10,12,9)(17,20,22,29)(18,24,26,32)(19,28,27,21)(23,31,30,25);(1,6)(2,5)(3,10)(4,8)(7,16)(9,14)(11,13)(12,15)(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,c^2=gi,d^2=i,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8653196);