#8080779

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

4478 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)(9,10,14,16)(11,17,20,15)(12,18,22,24)(13,19,23,21)(25,26)(27,31)(28,32)(29,39)(30,34)(33,37)(35,40)(36,38);(1,2,5,6)(3,7,4,8)(9,11)(10,15)(12,19)(13,18)(14,20)(16,17)(21,22)(23,24)(25,27)(26,31)(28,37)(29,36)(30,35)(32,33)(34,40)(38,39);(1,3,5,4)(2,7,6,8)(10,16)(13,23)(15,17)(18,24)(27,35)(29,37)(31,40)(33,39);(1,4,5,3)(2,8,6,7)(9,11)(10,17)(12,21)(13,18)(14,20)(15,16)(19,22)(23,24)(25,28)(26,32)(27,29)(30,36)(31,39)(33,40)(34,38)(35,37);(9,12,14,22)(10,18,16,24)(11,19,20,21)(13,17,23,15)(25,29)(26,33)(27,36)(28,35)(30,37)(31,32)(34,39)(38,40);(9,13,14,23)(10,19,16,21)(11,18,20,24)(12,17,22,15)(25,26)(27,31)(28,38)(29,33)(30,34)(32,36)(35,40)(37,39)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=gh,d^2=gh,e^2=h,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, 7987975);