#7726605

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

4558 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)(9,10,14,12)(11,17,18,15)(13,16,22,20)(19,24,21,23)(25,26)(27,37)(28,33)(29,34)(30,35)(31,36)(32,38)(39,40);(1,3,5,8)(2,6,4,7)(9,11)(10,15)(12,17)(13,21)(14,18)(16,24)(19,22)(20,23)(25,27,31,38)(26,32,36,37)(28,29,35,40)(30,34,33,39);(10,12)(11,18)(16,20)(19,21)(27,38)(29,40)(32,37)(34,39);(1,2,5,4)(3,6,8,7)(9,12,14,10)(11,17,18,15)(13,20,22,16)(19,24,21,23)(25,28,31,35)(26,33,36,30)(27,29,38,40)(32,39,37,34);(1,2,5,4)(3,6,8,7)(9,11)(10,15)(12,17)(13,19)(14,18)(16,23)(20,24)(21,22)(25,29)(26,34)(27,35)(28,38)(30,37)(31,40)(32,33)(36,39);(1,4,5,2)(3,7,8,6)(9,13,14,22)(10,16,12,20)(11,19,18,21)(15,23,17,24)(25,30,31,33)(26,35,36,28)(27,39,38,34)(29,37,40,32)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,d^2=g,e^2=gi,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,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, 7633801);