#10424588

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

4040 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:

(1,2,6,9)(3,4,11,7)(5,8,14,13)(10,12,16,15)(17,18)(19,25)(20,22)(21,28)(23,27)(24,30)(26,31)(29,32)(33,34)(35,36)(37,40)(38,41)(39,43)(42,44)(45,46)(47,48);(1,3,6,11)(2,7,9,4)(5,10,14,16)(8,15,13,12)(18,20)(23,25)(24,28)(31,32)(35,39)(36,43)(42,47)(44,48);(17,19)(18,23)(20,25)(21,26)(22,27)(24,31)(28,32)(29,30)(33,35,38,39)(34,36,41,43)(37,42,46,47)(40,44,45,48);(1,2)(3,7)(4,11)(5,13)(6,9)(8,14)(10,12)(15,16)(17,19)(18,23)(20,25)(21,29)(22,27)(24,32)(26,30)(28,31)(33,34)(35,36)(37,45)(38,41)(39,43)(40,46)(42,48)(44,47);(1,4,6,7)(2,3,9,11)(5,12,14,15)(8,10,13,16)(17,20,22,18)(19,25,27,23)(21,28,30,24)(26,32,29,31)(33,36)(34,39)(35,41)(37,44)(38,43)(40,47)(42,45)(46,48);(1,5)(2,8)(3,10)(4,12)(6,14)(7,15)(9,13)(11,16)(17,21,22,30)(18,24,20,28)(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=gh,b^2=gh,c^2=i,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=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, 10331784);