#8925012

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

4656 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)(4,6)(5,8)(9,10)(11,15)(12,16)(13,17)(14,19)(18,22)(20,23)(21,24)(25,26,31,35)(27,32,37,34)(28,38,36,33)(29,30,39,40);(1,2,4,6)(3,5,8,7)(9,11,13,19)(10,14,17,15)(12,18,20,24)(16,21,23,22)(25,27)(26,32)(28,39)(29,36)(30,33)(31,37)(34,35)(38,40);(1,2)(3,5)(4,6)(7,8)(12,20)(16,23)(18,24)(21,22)(25,28,31,36)(26,33,35,38)(27,29,37,39)(30,32,40,34);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,16)(13,17)(15,19)(18,21)(20,23)(22,24)(25,27)(26,34)(28,29)(30,33)(31,37)(32,35)(36,39)(38,40);(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23)(25,29)(26,30)(27,36)(28,37)(31,39)(32,33)(34,38)(35,40);(9,12)(10,16)(11,18)(13,20)(14,21)(15,22)(17,23)(19,24)(25,30)(26,29)(27,33)(28,34)(31,40)(32,36)(35,39)(37,38)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,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, 8832208);