#8533144

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	#53456	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3965 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=gi,d^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=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, 8440340);