#7643611

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	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4482 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^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, 7550807);