#10166664

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

4043 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)(3,4)(5,13)(6,7)(8,10)(9,15)(11,14)(12,16)(17,18,22,25)(19,23,28,31)(20,21,29,24)(26,27,32,30)(33,34,39,43)(35,40,37,42)(36,46,38,41)(44,48,45,47);(2,7)(4,8)(9,13)(12,14)(17,19,22,28)(18,23,25,31)(20,26,29,32)(21,27,24,30)(33,35,39,37)(34,40,43,42)(36,45,38,44)(41,48,46,47);(2,7)(3,10)(9,13)(11,16)(17,20,22,29)(18,24,25,21)(19,26,28,32)(23,30,31,27)(33,36,39,38)(34,41,43,46)(35,44,37,45)(40,47,42,48);(1,3,6,10)(2,8,7,4)(5,11,15,16)(9,14,13,12)(17,20)(18,21)(19,26)(22,29)(23,27)(24,25)(28,32)(30,31)(33,37,39,35)(34,42,43,40)(36,44,38,45)(41,47,46,48);(1,4,6,8)(2,3,7,10)(5,14,15,12)(9,16,13,11)(17,19)(18,23)(20,26)(21,27)(22,28)(24,30)(25,31)(29,32)(33,38,39,36)(34,41,43,46)(35,45,37,44)(40,47,42,48);(1,5,6,15)(2,9,7,13)(3,11,10,16)(4,12,8,14)(17,21)(18,20)(19,27)(22,24)(23,26)(25,29)(28,30)(31,32)(33,34)(35,40)(36,41)(37,42)(38,46)(39,43)(44,47)(45,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gh,d^2=gi,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10073860);