#7654009

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

4534 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7561205);