#7766572

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

4545 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)(11,12)(13,18)(14,16)(15,17)(19,20)(21,23)(22,24)(25,26,31,29)(27,35,36,32)(28,33,37,38)(30,34,40,39);(1,3,5,8)(2,6,4,7)(9,11)(10,15)(12,16)(13,19)(14,17)(18,22)(20,23)(21,24)(25,27)(26,32)(28,30)(29,35)(31,36)(33,39)(34,38)(37,40);(10,16)(11,17)(13,21)(20,22)(27,36)(30,40)(32,35)(34,39);(9,11)(10,12)(13,19)(14,17)(15,16)(18,20)(21,24)(22,23)(25,28)(26,33)(27,30)(29,38)(31,37)(32,39)(34,35)(36,40);(1,4,5,2)(3,7,8,6)(9,12)(10,17)(11,16)(13,20)(14,15)(18,24)(19,23)(21,22)(25,29,31,26)(27,35,36,32)(28,33,37,38)(30,39,40,34);(9,13)(10,18)(11,19)(12,20)(14,21)(15,22)(16,23)(17,24)(25,30)(26,34)(27,28)(29,39)(31,40)(32,38)(33,35)(36,37)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7673768);