#7826301

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

4556 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=h,d^2=gh,e^2=gh,f^2=gh,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, 7733497);