#8402299

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

4142 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)(4,9)(6,11)(7,8)(10,15)(12,14)(13,16)(33,34)(35,42)(36,41)(37,38)(39,40)(43,48)(44,45)(46,47);(1,3,7,10)(2,5,8,15)(4,12,13,6)(9,14,16,11)(17,18)(19,26)(20,21)(22,23)(24,29)(25,31)(27,28)(30,32)(34,40)(37,48)(41,42)(44,46);(2,8)(3,10)(6,12)(9,16)(18,23)(21,25)(24,26)(28,32)(33,35,39,36)(34,41,40,42)(37,44,48,46)(38,47,43,45);(1,4)(2,9)(3,12)(5,14)(6,10)(7,13)(8,16)(11,15)(17,19,22,29)(18,24,23,26)(20,30,31,27)(21,28,25,32)(33,36,39,35)(34,42,40,41)(37,44,48,46)(38,47,43,45);(1,5,7,15)(2,10,8,3)(4,11,13,14)(6,9,12,16)(17,20)(18,25)(19,27)(21,23)(22,31)(24,28)(26,32)(29,30)(33,37)(34,43)(35,44)(36,46)(38,40)(39,48)(41,45)(42,47);(1,6,7,12)(2,11,8,14)(3,4,10,13)(5,9,15,16)(17,21,22,25)(18,20,23,31)(19,28,29,32)(24,30,26,27)(33,38,39,43)(34,37,40,48)(35,45,36,47)(41,46,42,44)

Polycyclic presentation:

2^9|b^2=gh,c^2=g,d^2=hi,e^2=gh,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, 8309495);