#8087870

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	#2430	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4218 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)(9,10,14,13)(11,15,19,18)(12,16,22,21)(17,23,20,24)(25,26,30,28)(27,33,34,31)(29,32,37,36)(35,39,40,38);(1,2,5,6)(3,7,4,8)(25,27)(26,31)(28,33)(29,35)(30,34)(32,38)(36,39)(37,40);(10,13)(15,18)(16,21)(23,24)(27,34)(31,33)(35,40)(38,39);(1,3,5,4)(2,7,6,8)(9,11,14,19)(10,15,13,18)(12,20,22,17)(16,24,21,23)(25,28,30,26)(27,31,34,33)(29,32,37,36)(35,39,40,38);(9,12,14,22)(10,16,13,21)(11,17,19,20)(15,23,18,24)(25,27)(26,31)(28,33)(29,35)(30,34)(32,38)(36,39)(37,40);(1,4,5,3)(2,8,6,7)(9,13,14,10)(11,18,19,15)(12,21,22,16)(17,24,20,23)(25,29,30,37)(26,32,28,36)(27,35,34,40)(31,38,33,39)

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,d^2=gh,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, 7995066);