#8932985

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

4601 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8840181);