#7660625

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	#41957	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4300 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:

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

Polycyclic presentation:

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

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, 7567821);