#10412190

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	#45020	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4019 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=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, 10319386);