#7826268

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

4547 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=gh,d^2=h,e^2=gh,f^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,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, 7733464);