#7806767

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

4568 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=di,e^b=ei,f^b=fi,d^c=dh,f^c=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7713963);