#8425422

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	#2535	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3958 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=gi,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,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, 8332618);