#10420059

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	#58217	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4037 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)(17,18,23,25)(19,26,20,24)(21,22,31,32)(27,30,29,28)(33,34)(35,40)(36,38)(37,44)(39,43)(41,45)(42,46)(47,48);(1,2,5,6)(3,7,9,10)(4,8,13,14)(11,15,12,16)(33,35)(34,39)(36,40)(37,42)(38,43)(41,46)(44,47)(45,48);(17,19,23,20)(18,24,25,26)(21,27,31,29)(22,28,32,30)(33,34,38,36)(35,40,43,39)(37,41,45,44)(42,47,48,46);(1,3)(2,7)(4,12)(5,9)(6,10)(8,16)(11,13)(14,15)(17,20,23,19)(18,24,25,26)(21,27,31,29)(22,30,32,28)(33,36,38,34)(35,40,43,39)(37,44,45,41)(42,47,48,46);(1,2)(3,10)(4,8)(5,6)(7,9)(11,16)(12,15)(13,14)(17,21)(18,22)(19,27)(20,29)(23,31)(24,28)(25,32)(26,30)(33,35)(34,40)(36,39)(37,42)(38,43)(41,47)(44,46)(45,48);(1,4,5,13)(2,8,6,14)(3,11,9,12)(7,15,10,16)(17,22)(18,21)(19,28)(20,30)(23,32)(24,27)(25,31)(26,29)(33,37,38,45)(34,41,36,44)(35,42,43,48)(39,46,40,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=h,d^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, 10327255);