#8467215

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	#2540	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4020 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)(18,21)(20,24)(25,29)(28,30)(33,34,39,38)(35,40,45,43)(36,41,46,48)(37,47,44,42);(2,3)(7,9)(8,10)(14,15)(17,18,23,21)(19,24,27,20)(22,25,31,29)(26,30,32,28)(33,34,39,38)(35,43,45,40)(36,41,46,48)(37,42,44,47);(1,2,6,3)(4,7,12,9)(5,8,13,10)(11,14,16,15)(17,19,23,27)(18,24,21,20)(22,26,31,32)(25,30,29,28)(33,35,39,45)(34,40,38,43)(36,44,46,37)(41,42,48,47);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,20)(18,19)(21,27)(22,30)(23,24)(25,32)(26,29)(28,31)(33,36)(34,41)(35,44)(37,45)(38,48)(39,46)(40,42)(43,47);(1,4,6,12)(2,7,3,9)(5,11,13,16)(8,14,10,15)(17,21,23,18)(19,20,27,24)(22,29,31,25)(26,28,32,30)(33,37,39,44)(34,42,38,47)(35,36,45,46)(40,48,43,41);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,22)(18,25)(19,26)(20,28)(21,29)(23,31)(24,30)(27,32)(33,38,39,34)(35,43,45,40)(36,41,46,48)(37,47,44,42)

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=ghi,d^2=h,e^2=ghi,f^2=i,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, 8374411);