#8446015

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

4013 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=gh,d^2=h,e^2=gi,f^2=g,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, 8353211);