#8928010

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

4368 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)(3,5)(4,7)(6,8)(9,10)(11,14)(12,21)(13,17)(15,18)(16,20)(19,24)(22,23)(25,26,30,28)(27,33,35,31)(29,32,37,36)(34,39,40,38);(1,2,6,8)(3,5,7,4)(9,11)(10,14)(12,13)(15,19)(16,22)(17,21)(18,24)(20,23)(25,26,30,28)(27,31,35,33)(29,36,37,32)(34,39,40,38);(1,3,6,7)(2,4,8,5)(11,19)(12,20)(14,24)(16,21)(25,27)(26,31)(28,33)(29,34)(30,35)(32,38)(36,39)(37,40);(1,4,6,5)(2,3,8,7)(9,12)(10,16)(11,13)(14,22)(15,20)(17,24)(18,21)(19,23)(25,28,30,26)(27,33,35,31)(29,36,37,32)(34,39,40,38);(1,2)(3,4)(5,7)(6,8)(9,13)(10,17)(11,20)(12,19)(14,16)(15,23)(18,22)(21,24)(25,29,30,37)(26,32,28,36)(27,34,35,40)(31,38,33,39);(1,5,6,4)(2,7,8,3)(9,14)(10,11)(12,22)(13,16)(15,24)(17,20)(18,19)(21,23)(25,27)(26,31)(28,33)(29,34)(30,35)(32,38)(36,39)(37,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=g,d^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8835206);