#8418696

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	#36692	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3909 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=g,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8325892);