#7651843

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

4572 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)(9,10,14,13)(11,17,19,15)(12,16,22,21)(18,24,20,23)(25,26)(27,35)(28,32)(29,33)(30,34)(31,36)(37,38)(39,40);(1,2,4,5)(3,6,7,8)(25,27)(26,31)(28,29)(30,36)(32,40)(33,37)(34,35)(38,39);(10,13)(15,17)(16,21)(23,24)(27,36)(29,39)(31,35)(33,40);(9,11)(10,15)(12,20)(13,17)(14,19)(16,24)(18,22)(21,23)(25,28)(26,32)(27,29)(30,38)(31,40)(33,35)(34,37)(36,39);(1,3,4,7)(2,6,5,8)(9,12,14,22)(10,16,13,21)(11,18,19,20)(15,23,17,24)(25,26)(27,31)(28,37)(29,33)(30,34)(32,38)(35,36)(39,40);(9,13,14,10)(11,17,19,15)(12,21,22,16)(18,24,20,23)(25,29)(26,33)(27,28)(30,39)(31,37)(32,35)(34,40)(36,38)

Polycyclic presentation:

2^9|a^2=h,b^2=gi,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^c=fhi,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, 7559039);