#8908098

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

4571 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)(3,7)(4,8)(5,6)(9,10,14,15)(11,17,18,16)(12,13,21,23)(19,20,22,24)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,40)(37,38);(2,6)(3,8)(10,15)(12,21)(16,17)(19,22)(25,27,31,28)(26,32,30,33)(29,36,38,35)(34,40,37,39);(1,3)(2,4)(5,8)(6,7)(10,15)(11,18)(13,23)(19,22)(26,30)(27,28)(34,37)(35,36);(1,4)(2,3)(5,7)(6,8)(9,11)(10,16)(12,22)(13,20)(14,18)(15,17)(19,21)(23,24)(25,28,31,27)(26,33,30,32)(29,35,38,36)(34,39,37,40);(1,2)(3,4)(5,6)(7,8)(9,12)(10,13)(11,19)(14,21)(15,23)(16,24)(17,20)(18,22)(25,29,31,38)(26,34,30,37)(27,35,28,36)(32,39,33,40);(1,4)(2,3)(5,7)(6,8)(9,13)(10,12)(11,20)(14,23)(15,21)(16,22)(17,19)(18,24)(25,30,31,26)(27,33,28,32)(29,37,38,34)(35,40,36,39)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,d^2=gh,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8815294);