#7798046

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

4401 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=gh,d^2=i,e^2=i,f^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,e^b=ei,d^c=dh,f^c=fi,e^d=ei

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7705242);