#8193821

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3982 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,d^2=ghi,e^2=hi,f^2=g,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, 8101017);