#9783548

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	#2688	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3814 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=h,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,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, 9690744);