#8080993

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	C₂×D₄²	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4502 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,e^2=g,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^b=di,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, 7988189);