#7720402

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

4399 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)(9,10,14,11)(12,17,19,15)(13,18,21,16)(20,22,24,23)(25,26,31,28)(27,35,36,32)(29,33,38,39)(30,40,37,34);(1,2,5,6)(3,7,4,8)(9,10,14,11)(12,17,19,15)(13,16,21,18)(20,23,24,22)(25,27,31,36)(26,32,28,35)(29,30,38,37)(33,34,39,40);(10,11)(12,19)(16,18)(20,24)(27,36)(29,38)(32,35)(33,39);(1,3,5,4)(2,7,6,8)(9,11,14,10)(12,17,19,15)(13,18,21,16)(20,23,24,22)(25,28,31,26)(27,35,36,32)(29,39,38,33)(30,40,37,34);(1,4,5,3)(2,8,6,7)(9,12)(10,15)(11,17)(13,20)(14,19)(16,22)(18,23)(21,24)(25,29,31,38)(26,33,28,39)(27,37,36,30)(32,40,35,34);(9,13,14,21)(10,16,11,18)(12,20,19,24)(15,22,17,23)(25,30,31,37)(26,34,28,40)(27,38,36,29)(32,39,35,33)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,d^2=gh,e^2=g,f^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=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, 7627598);