#10445000

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

4187 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=hi,d^2=gi,e^2=gi,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, 10352196);