#8193897

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

4154 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=g,d^2=gi,e^2=i,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, 8101093);