#8188014

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

4134 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)(3,12)(4,7)(5,15)(6,10)(8,11)(9,14)(13,16)(17,18,23,22)(19,26,21,24)(20,25,28,30)(27,32,29,31)(33,34)(35,36)(37,40)(38,45)(39,41)(42,47)(43,48)(44,46);(1,3,8,14)(2,9,11,12)(4,10,5,16)(6,7,13,15)(20,28)(25,30)(27,29)(31,32)(33,35,40,39)(34,41,37,36)(38,43,47,46)(42,48,45,44);(1,4,8,5)(2,7,11,15)(3,10,14,16)(6,9,13,12)(17,19,23,21)(18,24,22,26)(20,27,28,29)(25,31,30,32)(33,36)(34,39)(35,37)(38,44)(40,41)(42,46)(43,45)(47,48);(1,5,8,4)(2,7,11,15)(3,10,14,16)(6,12,13,9)(17,20,23,28)(18,25,22,30)(19,27,21,29)(24,31,26,32)(33,37,40,34)(35,41,39,36)(38,45,47,42)(43,48,46,44);(1,6,8,13)(2,10,11,16)(3,7,14,15)(4,9,5,12)(17,21,23,19)(18,26,22,24)(20,29,28,27)(25,32,30,31)(33,38)(34,42)(35,43)(36,44)(37,45)(39,46)(40,47)(41,48);(1,7)(2,4)(3,13)(5,11)(6,14)(8,15)(9,10)(12,16)(17,22,23,18)(19,26,21,24)(20,30,28,25)(27,32,29,31)(33,39,40,35)(34,41,37,36)(38,46,47,43)(42,48,45,44)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=gh,d^2=ghi,e^2=gh,f^2=h,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, 8095210);