#8069754

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4464 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,e^a=ei,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, 7976950);