#8402606

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3971 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)(17,18,23,20)(19,24,27,29)(21,25,31,28)(22,26,32,30)(33,34,39,41)(35,37,36,43)(38,40,46,48)(42,45,44,47);(1,3,7,10)(2,5,8,15)(4,12,13,6)(9,14,16,11)(17,19,23,27)(18,24,20,29)(21,30,31,26)(22,28,32,25)(33,34)(35,37)(36,43)(38,40)(39,41)(42,45)(44,47)(46,48);(2,8)(3,10)(6,12)(9,16)(19,27)(22,32)(24,29)(26,30)(33,35,39,36)(34,37,41,43)(38,44,46,42)(40,47,48,45);(1,4)(2,9)(3,12)(5,14)(6,10)(7,13)(8,16)(11,15)(17,20,23,18)(19,24,27,29)(21,25,31,28)(22,30,32,26)(33,36,39,35)(34,37,41,43)(38,44,46,42)(40,45,48,47);(1,5,7,15)(2,10,8,3)(4,11,13,14)(6,9,12,16)(17,21)(18,25)(19,26)(20,28)(22,29)(23,31)(24,32)(27,30)(33,37)(34,36)(35,41)(38,45)(39,43)(40,44)(42,48)(46,47);(1,6,7,12)(2,11,8,14)(3,4,10,13)(5,9,15,16)(17,22,23,32)(18,26,20,30)(19,28,27,25)(21,29,31,24)(33,38,39,46)(34,40,41,48)(35,42,36,44)(37,45,43,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=g,d^2=ghi,e^2=h,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, 8309802);