#8087991

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

4041 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=gh,d^2=h,e^2=h,f^2=gh,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, 7995187);