#7647124

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	#42970	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4303 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)(10,16)(14,18)(17,20)(22,24)(25,26,30,33)(27,34,28,31)(29,37,38,32)(35,39,36,40);(1,2)(3,6)(4,5)(7,8)(9,10)(11,20)(12,17)(13,18)(14,19)(15,16)(21,22)(23,24);(9,11,15,12)(10,17,16,20)(13,21,19,23)(14,22,18,24)(25,27,30,28)(26,31,33,34)(29,35,38,36)(32,39,37,40);(9,12,15,11)(10,17,16,20)(13,21,19,23)(14,24,18,22)(25,26)(27,31)(28,34)(29,32)(30,33)(35,39)(36,40)(37,38);(9,13,15,19)(10,18,16,14)(11,21,12,23)(17,24,20,22)(25,28,30,27)(26,31,33,34)(29,36,38,35)(32,39,37,40);(1,3,4,7)(2,6,5,8)(9,14,15,18)(10,19,16,13)(11,22,12,24)(17,23,20,21)(25,29)(26,32)(27,35)(28,36)(30,38)(31,39)(33,37)(34,40)

Polycyclic presentation:

2^9|a^2=h,c^2=hi,d^2=i,e^2=hi,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, 7554320);