#8406467

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	#2678	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3845 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)(3,8)(4,5)(6,7)(9,13)(10,16)(11,14)(12,15)(17,18,21,22)(19,25,27,23)(20,24,28,29)(26,31,32,30)(34,39)(36,42)(40,46)(44,48);(2,7)(4,13)(8,15)(10,14)(18,22)(23,25)(24,29)(30,31)(35,43)(40,46)(44,48)(45,47);(2,7)(5,9)(8,15)(11,16)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(33,34)(35,40)(36,41)(37,42)(38,39)(43,46)(44,47)(45,48);(1,3)(2,8)(4,14)(5,11)(6,12)(7,15)(9,16)(10,13)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(33,35,38,43)(34,40,39,46)(36,44,42,48)(37,45,41,47);(1,4)(2,9)(3,10)(5,7)(6,13)(8,11)(12,14)(15,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31)(33,36,38,42)(34,41,39,37)(35,44,43,48)(40,47,46,45);(1,5,6,9)(2,4,7,13)(3,11,12,16)(8,14,15,10)(17,20,21,28)(18,24,22,29)(19,26,27,32)(23,30,25,31)(33,37,38,41)(34,42,39,36)(35,45,43,47)(40,48,46,44)

Polycyclic presentation:

2^9|a^2=i,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, 8313663);