#8425456

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4203 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=hi,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, 8332652);