#7651144

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	#39793	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4243 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=h,d^2=g,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^c=fhi,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, 7558340);