#7726778

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
prime radical	#2403	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4595 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,d^2=g,e^2=hi,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, 7633974);