#9712155

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

3863 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=gi,d^2=gh,e^2=g,f^2=hi,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, 9619351);