#7826306

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

4412 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=gh,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, 7733502);