#7644110

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4096 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7551306);