#8406472

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	#2812	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3855 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=g,d^2=gh,e^2=gh,f^2=h,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, 8313668);