#8464256

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

4151 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=ghi,d^2=h,e^2=gi,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, 8371452);