#8661368

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

4147 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8568564);