#7645382

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	#35236	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4542 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)(11,18)(12,17)(13,14)(15,19)(16,22)(20,24)(21,23)(25,26,31,33)(27,30,29,39)(28,35,38,32)(34,40,36,37);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,23)(13,20)(14,21)(15,22)(17,24)(18,19)(25,27,31,29)(26,30,33,39)(28,34,38,36)(32,37,35,40);(9,12,15,13)(10,17,19,14)(11,20,22,23)(16,21,18,24)(25,28)(26,32)(27,34)(29,36)(30,37)(31,38)(33,35)(39,40);(9,11)(10,18)(12,20)(13,23)(14,21)(15,22)(16,19)(17,24)(25,29,31,27)(26,30,33,39)(28,36,38,34)(32,37,35,40);(1,3,4,7)(2,6,5,8)(9,13,15,12)(10,14,19,17)(11,20,22,23)(16,21,18,24)(25,30)(26,29)(27,33)(28,37)(31,39)(32,36)(34,35)(38,40);(9,14)(10,12)(11,21)(13,19)(15,17)(16,23)(18,20)(22,24)(25,28)(26,32)(27,34)(29,36)(30,37)(31,38)(33,35)(39,40)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=i,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^d=ehi

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, 7552578);