#8750137

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	#2414	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3943 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)(3,4)(5,9)(6,7)(8,12)(10,11)(13,14)(15,16)(17,18,23,27)(19,24,22,26)(20,25,31,21)(28,30,29,32)(33,34,39,43)(35,41,37,40)(36,42,47,38)(44,45,46,48);(2,7)(3,8)(9,16)(10,13)(17,19,23,22)(18,24,27,26)(20,29,31,28)(21,30,25,32)(33,35,39,37)(34,40,43,41)(36,44,47,46)(38,45,42,48);(2,7)(3,8)(9,16)(10,13)(17,20,23,31)(18,25,27,21)(19,28,22,29)(24,30,26,32)(33,35,39,37)(34,41,43,40)(36,46,47,44)(38,45,42,48);(1,3,6,8)(2,4,7,12)(5,13,15,10)(9,14,16,11)(17,20)(18,21)(19,29)(22,28)(23,31)(24,30)(25,27)(26,32)(33,36,39,47)(34,42,43,38)(35,44,37,46)(40,48,41,45);(1,4)(2,8)(3,7)(5,14)(6,12)(9,10)(11,15)(13,16)(17,21)(18,20)(19,30)(22,32)(23,25)(24,29)(26,28)(27,31)(33,37,39,35)(34,40,43,41)(36,46,47,44)(38,45,42,48);(1,5,6,15)(2,9,7,16)(3,10,8,13)(4,11,12,14)(17,22,23,19)(18,26,27,24)(20,29,31,28)(21,30,25,32)(33,38,39,42)(34,36,43,47)(35,45,37,48)(40,44,41,46)

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=ghi,d^2=g,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8657333);