#9712167

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

3865 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=ghi,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, 9619363);