#7715930

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

4474 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)(9,10,14,15)(11,13,17,23)(12,19,21,16)(18,24,20,22)(26,31)(29,33)(32,38)(35,39);(1,2)(3,7)(4,8)(5,6)(9,11)(10,13)(12,20)(14,17)(15,23)(16,24)(18,21)(19,22)(25,26)(27,32)(28,29)(30,31)(33,37)(34,35)(36,38)(39,40);(10,15)(11,17)(12,21)(22,24)(26,31)(28,37)(32,38)(34,40);(1,3,5,4)(2,7,6,8)(9,11)(10,13)(12,18)(14,17)(15,23)(16,22)(19,24)(20,21)(25,27,30,36)(26,32,31,38)(28,34,37,40)(29,35,33,39);(9,12,14,21)(10,16,15,19)(11,18,17,20)(13,22,23,24)(25,28)(26,33)(27,34)(29,31)(30,37)(32,39)(35,38)(36,40);(1,4,5,3)(2,8,6,7)(9,13)(10,11)(12,22)(14,23)(15,17)(16,18)(19,20)(21,24)(25,29,30,33)(26,28,31,37)(27,35,36,39)(32,34,38,40)

Polycyclic presentation:

2^9|a^2=h,d^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, 7623126);