#9712227

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	#39572	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3950 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)(3,11)(4,9)(5,13)(6,7)(8,12)(10,15)(14,16)(17,18)(19,22)(20,27)(21,24)(23,28)(25,31)(26,30)(29,32)(33,34,39,43)(35,40,38,42)(36,37,41,47)(44,48,46,45);(2,7)(4,14)(8,11)(10,13)(17,19,21,26)(18,22,24,30)(20,25,28,32)(23,29,27,31)(33,35,39,38)(34,40,43,42)(36,46,41,44)(37,45,47,48);(2,7)(3,12)(9,16)(10,13)(17,18,21,24)(19,22,26,30)(20,23,28,27)(25,29,32,31)(33,36,39,41)(34,37,43,47)(35,44,38,46)(40,48,42,45);(1,3,6,12)(2,8,7,11)(4,13,14,10)(5,9,15,16)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,28)(31,32);(1,4,6,14)(2,9,7,16)(3,13,12,10)(5,8,15,11)(17,19)(18,22)(20,25)(21,26)(23,29)(24,30)(27,31)(28,32)(33,37)(34,41)(35,45)(36,43)(38,48)(39,47)(40,44)(42,46);(1,5,6,15)(2,10,7,13)(3,9,12,16)(4,11,14,8)(17,20)(18,23)(19,25)(21,28)(22,29)(24,27)(26,32)(30,31)(33,38,39,35)(34,42,43,40)(36,46,41,44)(37,45,47,48)

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=ghi,d^2=g,e^2=g,f^2=gi,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, 9619423);