#7646658

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4213 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)(17,19)(21,24)(25,26,30,29)(27,33,35,31)(28,36,37,32)(34,38,40,39);(1,2)(3,6)(4,5)(7,8)(9,10,15,16)(11,19,13,17)(12,18,22,14)(20,21,23,24);(9,11,15,13)(10,17,16,19)(12,20,22,23)(14,21,18,24)(25,27)(26,31)(28,34)(29,33)(30,35)(32,38)(36,39)(37,40);(9,12)(10,18)(11,20)(13,23)(14,16)(15,22)(17,24)(19,21)(25,28,30,37)(26,32,29,36)(27,34,35,40)(31,38,33,39);(9,13,15,11)(10,17,16,19)(12,20,22,23)(14,24,18,21)(25,29,30,26)(27,33,35,31)(28,32,37,36)(34,38,40,39);(1,3,4,7)(2,6,5,8)(9,14,15,18)(10,12,16,22)(11,21,13,24)(17,20,19,23)(25,27)(26,31)(28,34)(29,33)(30,35)(32,38)(36,39)(37,40)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=i,d^2=h,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, 7553854);