#8188031

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	#36332	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4052 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)(3,10)(4,7)(5,8)(6,9)(11,16)(12,14)(13,15)(17,18,23,22)(19,26,21,24)(20,25,28,30)(27,32,29,31)(34,38)(35,41)(40,43)(42,47);(1,3,7,5)(2,8,4,10)(6,11,14,13)(9,15,12,16)(20,28)(25,30)(27,29)(31,32)(34,38)(39,45)(40,43)(46,48);(6,14)(9,12)(11,13)(15,16)(17,19,23,21)(18,24,22,26)(20,27,28,29)(25,31,30,32)(33,34,37,38)(35,39,41,45)(36,43,44,40)(42,48,47,46);(1,4,7,2)(3,8,5,10)(6,12,14,9)(11,15,13,16)(17,20,23,28)(18,25,22,30)(19,27,21,29)(24,31,26,32);(1,5,7,3)(2,8,4,10)(6,13,14,11)(9,15,12,16)(17,21,23,19)(18,26,22,24)(20,29,28,27)(25,32,30,31)(33,35,37,41)(34,39,38,45)(36,42,44,47)(40,46,43,48);(1,6,7,14)(2,9,4,12)(3,11,5,13)(8,15,10,16)(17,22,23,18)(19,26,21,24)(20,30,28,25)(27,32,29,31)(33,36,37,44)(34,40,38,43)(35,42,41,47)(39,46,45,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=h,d^2=ghi,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8095227);