#7644319

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	#42928	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4224 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)(9,10,15,12)(11,18,13,16)(14,20,22,17)(19,23,21,24)(27,28)(30,38)(32,35)(33,37);(1,3,5,8)(2,6,4,7)(9,11,15,13)(10,16,12,18)(14,19,22,21)(17,23,20,24)(25,26,31,29)(27,32,28,35)(30,33,38,37)(34,39,36,40);(11,13)(16,18)(19,21)(23,24)(25,27,31,28)(26,32,29,35)(30,34,38,36)(33,39,37,40);(9,12,15,10)(11,16,13,18)(14,20,22,17)(19,23,21,24)(25,28,31,27)(26,32,29,35)(30,36,38,34)(33,39,37,40);(9,13,15,11)(10,16,12,18)(14,21,22,19)(17,23,20,24)(25,29,31,26)(27,32,28,35)(30,37,38,33)(34,39,36,40);(1,4,5,2)(3,7,8,6)(9,14,15,22)(10,17,12,20)(11,19,13,21)(16,23,18,24)(25,30)(26,33)(27,34)(28,36)(29,37)(31,38)(32,39)(35,40)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,d^2=hi,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, 7551515);