#8402441

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	#45608	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4138 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)(3,5)(4,12)(6,8)(7,11)(9,15)(10,16)(13,14)(17,18,23,20)(19,24,27,29)(21,25,32,30)(22,26,28,31)(33,34)(35,40)(36,38)(37,41)(39,43)(42,46)(44,48)(45,47);(1,3)(2,7)(4,13)(5,8)(6,11)(9,14)(10,15)(12,16)(17,19)(18,24)(20,29)(21,22)(23,27)(25,26)(28,32)(30,31)(33,35,39,46)(34,40,43,42)(36,47,37,44)(38,45,41,48);(2,8)(5,7)(9,12)(14,16)(19,27)(22,28)(24,29)(26,31)(33,36,39,37)(34,41,43,38)(35,44,46,47)(40,45,42,48);(1,3)(2,7)(4,13)(5,8)(6,11)(9,14)(10,15)(12,16)(17,20,23,18)(19,24,27,29)(21,25,32,30)(22,31,28,26)(33,37,39,36)(34,41,43,38)(35,44,46,47)(40,48,42,45);(1,4,6,15)(2,9,8,12)(3,10,11,13)(5,14,7,16)(17,21)(18,25)(19,28)(20,30)(22,27)(23,32)(24,31)(26,29)(33,35)(34,42)(36,44)(37,47)(38,45)(39,46)(40,43)(41,48);(1,5,6,7)(2,3,8,11)(4,14,15,16)(9,10,12,13)(17,22,23,28)(18,26,20,31)(19,21,27,32)(24,25,29,30)(33,38,39,41)(34,36,43,37)(35,45,46,48)(40,47,42,44)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=gh,d^2=i,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8309637);