#8828645

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4184 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)(6,7)(8,10)(9,13)(11,16)(14,15)(18,20)(19,25)(24,28)(26,31)(33,34,38,41)(35,39,42,36)(37,40,46,48)(43,47,45,44);(2,7)(4,13)(5,8)(11,14)(17,18,22,20)(19,23,25,27)(21,28,29,24)(26,32,31,30)(33,35,38,42)(34,39,41,36)(37,45,46,43)(40,44,48,47);(2,7)(3,10)(9,12)(11,14)(19,25)(23,27)(26,31)(30,32)(35,42)(36,39)(43,45)(44,47);(1,3,6,10)(2,8,7,5)(4,14,13,11)(9,15,12,16)(17,19,22,25)(18,23,20,27)(21,26,29,31)(24,30,28,32)(33,36)(34,35)(37,44)(38,39)(40,43)(41,42)(45,48)(46,47);(1,4,6,13)(2,9,7,12)(3,11,10,14)(5,15,8,16)(17,20,22,18)(19,23,25,27)(21,28,29,24)(26,30,31,32)(33,34)(35,39)(36,42)(37,40)(38,41)(43,47)(44,45)(46,48);(1,5,6,8)(2,10,7,3)(4,15,13,16)(9,14,12,11)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)(33,37)(34,40)(35,43)(36,44)(38,46)(39,47)(41,48)(42,45)

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,d^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

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, 8735841);