#8069529

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
prime radical	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4210 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gi,d^2=h,e^2=h,f^2=i,b^a=bg,c^a=ch,d^a=di,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7976725);