#5451019

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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1712 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:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=fi,d^2=fgh,e^2=gi,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, 5358215);