#8504951

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	#42533	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3958 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)(3,9)(4,12)(5,14)(6,11)(7,8)(10,15)(13,16)(17,18,23,20)(19,24,29,27)(21,30,32,25)(22,26,28,31)(33,34)(35,39)(36,42)(37,45)(38,41)(40,46)(43,48)(44,47);(2,8)(5,15)(9,12)(11,13)(17,19)(18,24)(20,27)(21,22)(23,29)(25,31)(26,30)(28,32)(33,35,38,36)(34,39,41,42)(37,43,46,44)(40,47,45,48);(1,3,7,4)(2,9,8,12)(5,11,15,13)(6,10,16,14)(17,18,23,20)(19,27,29,24)(21,25,32,30)(22,26,28,31)(33,36,38,35)(34,39,41,42)(37,44,46,43)(40,47,45,48);(1,4,7,3)(2,9,8,12)(5,13,15,11)(6,10,16,14)(17,20,23,18)(19,27,29,24)(21,30,32,25)(22,26,28,31)(33,34)(35,42)(36,39)(37,40)(38,41)(43,48)(44,47)(45,46);(1,5,7,15)(2,10,8,14)(3,11,4,13)(6,9,16,12)(17,21)(18,25)(19,28)(20,30)(22,29)(23,32)(24,26)(27,31)(33,37)(34,40)(35,43)(36,44)(38,46)(39,47)(41,45)(42,48);(1,6,7,16)(2,11,8,13)(3,10,4,14)(5,9,15,12)(17,22,23,28)(18,26,20,31)(19,21,29,32)(24,30,27,25)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=ghi,d^2=gi,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8412147);