#8928012

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

4346 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)(4,8)(6,7)(9,10)(11,14)(12,21)(13,17)(15,18)(16,20)(19,24)(22,23)(25,26,31,35)(27,36,40,32)(28,33,38,30)(29,34,37,39);(2,7)(3,8)(9,11)(10,14)(12,13)(15,19)(16,22)(17,21)(18,24)(20,23)(25,27,31,40)(26,32,35,36)(28,37,38,29)(30,39,33,34);(1,3,6,8)(2,4,7,5)(11,19)(12,20)(14,24)(16,21)(25,28,31,38)(26,33,35,30)(27,37,40,29)(32,34,36,39);(1,4,6,5)(2,3,7,8)(9,12)(10,16)(11,13)(14,22)(15,20)(17,24)(18,21)(19,23)(25,28)(26,33)(27,37)(29,40)(30,35)(31,38)(32,34)(36,39);(1,2)(3,4)(5,8)(6,7)(9,13)(10,17)(11,20)(12,19)(14,16)(15,23)(18,22)(21,24)(25,29,31,37)(26,34,35,39)(27,38,40,28)(30,32,33,36);(1,5,6,4)(2,8,7,3)(9,14)(10,11)(12,22)(13,16)(15,24)(17,20)(18,19)(21,23)(25,30,31,33)(26,28,35,38)(27,39,40,34)(29,32,37,36)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,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, 8835208);