#8087929

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	#2676	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4117 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)(9,10,15,19)(11,17,13,16)(12,14,18,24)(20,21,22,23)(25,26)(27,30)(28,32)(29,33)(31,34)(35,37)(36,39)(38,40);(1,2)(3,6)(4,5)(7,8)(9,11,15,13)(10,16,19,17)(12,22,18,20)(14,21,24,23);(11,13)(16,17)(20,22)(21,23)(25,27,31,28)(26,32,34,30)(29,35,38,36)(33,39,40,37);(9,12)(10,14)(11,20)(13,22)(15,18)(16,23)(17,21)(19,24)(25,28,31,27)(26,32,34,30)(29,35,38,36)(33,37,40,39);(1,3,4,7)(2,6,5,8)(9,13,15,11)(10,17,19,16)(12,20,18,22)(14,23,24,21)(25,29,31,38)(26,33,34,40)(27,35,28,36)(30,37,32,39);(9,14)(10,18)(11,21)(12,19)(13,23)(15,24)(16,20)(17,22)(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=i,c^2=h,d^2=h,e^2=gh,f^2=h,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, 7995125);