#771639

order	512
nilpotency class	2
isoclinism	isoclinic class of #763332
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1670 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=h,c^2=hi,d^2=i,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=df,e^c=ef

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, 678835);