#7645416

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

4588 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)(9,10,14,18)(11,16,12,15)(13,17,21,24)(19,23,20,22)(25,26)(27,34)(28,33)(29,30)(31,35)(32,38)(36,40)(37,39);(1,2)(3,6)(4,5)(7,8)(25,27)(26,32)(28,39)(29,36)(30,37)(31,38)(33,40)(34,35);(9,11,14,12)(10,15,18,16)(13,19,21,20)(17,22,24,23)(25,28,31,29)(26,33,35,30)(27,36,38,39)(32,37,34,40);(9,12,14,11)(10,16,18,15)(13,19,21,20)(17,22,24,23)(25,27)(26,34)(28,36)(29,39)(30,37)(31,38)(32,35)(33,40);(1,3,4,7)(2,6,5,8)(9,13)(10,17)(11,19)(12,20)(14,21)(15,22)(16,23)(18,24)(25,29,31,28)(26,30,35,33)(27,36,38,39)(32,37,34,40);(9,10)(11,15)(12,16)(13,17)(14,18)(19,22)(20,23)(21,24)(25,30)(26,28)(27,37)(29,35)(31,33)(32,39)(34,36)(38,40)

Polycyclic presentation:

2^9|a^2=h,c^2=hi,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, 7552612);