#7646778

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	#41663	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4228 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|c^2=h,d^2=i,e^2=i,f^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, 7553974);