#9712088

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	#2414	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3929 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)(3,5)(4,8)(6,7)(9,13)(10,12)(11,14)(15,16)(17,18,23,21)(19,25,20,24)(22,26,30,29)(27,32,28,31)(33,34)(35,41)(36,44)(37,43)(38,42)(39,40)(45,48)(46,47);(4,11)(8,14)(10,15)(12,16)(17,19,23,20)(18,24,21,25)(22,27,30,28)(26,31,29,32)(33,35,40,47)(34,41,39,46)(36,45,38,43)(37,44,48,42);(2,7)(3,9)(8,14)(10,15)(33,36,40,38)(34,42,39,44)(35,43,47,45)(37,41,48,46);(1,3,6,9)(2,5,7,13)(4,10,11,15)(8,12,14,16)(17,20,23,19)(18,25,21,24)(22,28,30,27)(26,32,29,31)(33,37,40,48)(34,43,39,45)(35,44,47,42)(36,46,38,41);(1,4,6,11)(2,8,7,14)(3,10,9,15)(5,12,13,16)(17,21,23,18)(19,25,20,24)(22,26,30,29)(27,31,28,32)(33,38,40,36)(34,42,39,44)(35,45,47,43)(37,41,48,46);(1,5)(2,9)(3,7)(4,12)(6,13)(8,15)(10,14)(11,16)(17,22)(18,26)(19,27)(20,28)(21,29)(23,30)(24,31)(25,32)(33,39,40,34)(35,46,47,41)(36,44,38,42)(37,45,48,43)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=i,d^2=gh,e^2=gh,f^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=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, 9619284);