#8295562

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	#40649	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3932 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8202758);