#8718039

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

4047 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)(3,6)(4,9)(5,8)(7,12)(10,14)(11,16)(13,15)(17,18)(19,20)(21,25)(22,23)(24,26)(27,28)(29,30)(31,32)(33,34,39,43)(35,40,44,47)(36,41,45,46)(37,38,42,48);(2,3)(7,9)(8,10)(15,16)(17,18)(19,26)(20,24)(21,25)(22,23)(27,30)(28,29)(31,32)(33,35,39,44)(34,40,43,47)(36,37,45,42)(38,46,48,41);(2,3)(7,9)(8,10)(15,16)(18,23)(19,24)(25,32)(27,29)(35,44)(36,45)(40,47)(41,46);(1,3,6,2)(4,9,12,7)(5,8,14,10)(11,15,13,16)(17,19,22,24)(18,20,23,26)(21,29,31,27)(25,30,32,28)(33,36,39,45)(34,41,43,46)(35,37,44,42)(38,47,48,40);(1,4,6,12)(2,7,3,9)(5,13,14,11)(8,16,10,15)(17,20)(18,24)(19,23)(21,30)(22,26)(25,27)(28,31)(29,32)(33,37)(34,38)(35,45)(36,44)(39,42)(40,46)(41,47)(43,48);(1,5,6,14)(2,8,3,10)(4,11,12,13)(7,15,9,16)(17,21,22,31)(18,25,23,32)(19,27,24,29)(20,28,26,30)(33,38)(34,42)(35,46)(36,40)(37,43)(39,48)(41,44)(45,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,d^2=gi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8625235);