#9058392

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	#53430	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4149 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8965588);