#9015458

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	#53545	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3910 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)(3,7,9,11)(5,12,13,8)(10,16,15,14)(17,18,23,28)(19,26,21,24)(20,30,31,25)(22,27,29,32)(33,34,39,37)(35,40,45,43)(36,41,47,38)(42,44,46,48);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,16)(13,15)(17,19,23,21)(18,24,28,26)(20,22,31,29)(25,27,30,32)(33,35)(34,40)(36,46)(37,43)(38,44)(39,45)(41,48)(42,47);(2,4)(3,9)(5,13)(14,16)(17,20,23,31)(18,25,28,30)(19,29,21,22)(24,32,26,27)(33,36,39,47)(34,41,37,38)(35,42,45,46)(40,44,43,48);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,16)(13,15)(17,21,23,19)(18,26,28,24)(20,22,31,29)(25,27,30,32)(33,37,39,34)(35,43,45,40)(36,41,47,38)(42,44,46,48);(1,4,6,2)(3,7,9,11)(5,12,13,8)(10,14,15,16)(17,22)(18,27)(19,20)(21,31)(23,29)(24,25)(26,30)(28,32)(33,38,39,41)(34,36,37,47)(35,44,45,48)(40,46,43,42);(1,5,6,13)(2,8,4,12)(3,10,9,15)(7,14,11,16)(17,18)(19,24)(20,25)(21,26)(22,27)(23,28)(29,32)(30,31)(33,35)(34,40)(36,46)(37,43)(38,44)(39,45)(41,48)(42,47)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=ghi,d^2=ghi,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8922654);