#10423051

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	#59689	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4034 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=i,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, 10330247);