#7715929

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

4524 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)(9,10,14,15)(11,13,17,23)(12,19,21,16)(18,24,20,22)(25,26)(27,33)(28,31)(29,34)(30,32)(35,40)(36,38)(37,39);(1,3,5,8)(2,6,4,7)(9,11)(10,13)(12,20)(14,17)(15,23)(16,24)(18,21)(19,22)(25,27,30,28)(26,31,32,33)(29,35,37,36)(34,38,39,40);(10,15)(11,17)(12,21)(22,24)(26,32)(31,33)(34,39)(38,40);(9,11)(10,13)(12,18)(14,17)(15,23)(16,22)(19,24)(20,21)(25,26)(27,31)(28,33)(29,34)(30,32)(35,38)(36,40)(37,39);(9,12,14,21)(10,16,15,19)(11,18,17,20)(13,22,23,24)(25,28,30,27)(26,33,32,31)(29,36,37,35)(34,40,39,38);(1,4,5,2)(3,7,8,6)(9,13)(10,11)(12,22)(14,23)(15,17)(16,18)(19,20)(21,24)(25,29,30,37)(26,34,32,39)(27,35,28,36)(31,38,33,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,e^2=h,f^2=g,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, 7623125);