#8258628

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

4045 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=i,d^2=gh,e^2=ghi,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8165824);