#7798096

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

4364 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:

(1,2,5,4)(3,7,8,6)(9,10,15,17)(11,19,20,16)(12,18,22,21)(13,23,24,14)(25,26,31,28)(27,32,34,30)(29,33,39,36)(35,40,37,38);(1,3,5,8)(2,6,4,7)(9,11,15,20)(10,16,17,19)(12,13,22,24)(14,21,23,18)(25,27,31,34)(26,32,28,30)(29,37,39,35)(33,38,36,40);(1,2,5,4)(3,6,8,7)(10,17)(11,20)(12,22)(14,23)(27,34)(30,32)(35,37)(38,40);(9,12,15,22)(10,18,17,21)(11,13,20,24)(14,19,23,16)(25,28,31,26)(27,30,34,32)(29,33,39,36)(35,40,37,38);(1,2,5,4)(3,6,8,7)(9,13)(10,14)(11,12)(15,24)(16,18)(17,23)(19,21)(20,22)(25,29,31,39)(26,33,28,36)(27,35,34,37)(30,38,32,40);(1,4,5,2)(3,7,8,6)(9,14)(10,13)(11,21)(12,19)(15,23)(16,22)(17,24)(18,20)(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=ghi,c^2=gh,d^2=i,e^2=gi,f^2=gh,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, 7705292);