#5331038

order	512
nilpotency class	2
isoclinism	isoclinic class of #5211343
automorphism	Aut(G) order 1048576
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

1654 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fg,c^2=fhi,d^2=fi,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5238234);