#8418249

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	#2433	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4087 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)(3,11)(4,9)(5,10)(6,8)(7,14)(12,16)(13,15)(17,18)(19,24)(20,30)(21,25)(22,26)(23,27)(28,32)(29,31)(35,36)(37,45)(40,42)(41,46);(1,3,6,14)(2,7,8,11)(4,12,15,10)(5,13,16,9)(17,19)(18,24)(20,22)(21,31)(23,28)(25,29)(26,30)(27,32)(33,34,39,38)(35,42,36,40)(37,46,45,41)(43,47,44,48);(2,8)(5,16)(7,11)(9,13)(19,28)(22,29)(24,32)(26,31)(33,35,39,36)(34,40,38,42)(37,43,45,44)(41,47,46,48);(1,4,6,15)(2,9,8,13)(3,12,14,10)(5,11,16,7)(17,20)(18,21)(19,29)(22,28)(23,25)(24,26)(27,30)(31,32)(33,36,39,35)(34,40,38,42)(37,43,45,44)(41,48,46,47);(1,3)(2,7)(4,10)(5,13)(6,14)(8,11)(9,16)(12,15)(17,21)(18,25)(19,26)(20,27)(22,24)(23,30)(28,31)(29,32)(33,37,39,45)(34,41,38,46)(35,43,36,44)(40,47,42,48);(1,5,6,16)(2,10,8,12)(3,13,14,9)(4,11,15,7)(17,22,23,29)(18,26,27,31)(19,20,28,25)(21,24,30,32)(33,38,39,34)(35,40,36,42)(37,46,45,41)(43,47,44,48)

Polycyclic presentation:

2^9|b^2=h,c^2=gh,d^2=h,e^2=gh,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, 8325445);