#5751633

order	512
nilpotency class	2
isoclinism	isoclinic class of #5605735
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

1793 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)(10,12)(11,15)(17,18,23,24)(19,20,21,22)(25,26)(27,35)(28,33)(29,34)(30,31)(32,36)(37,38)(39,40);(4,8)(6,7)(10,12)(14,16)(17,19,23,21)(18,20,24,22)(26,31)(27,36)(29,33)(37,40);(2,3)(6,7)(9,10,13,12)(11,16,15,14)(17,20,23,22)(18,19,24,21)(25,27)(26,32)(28,37)(29,38)(30,36)(31,35)(33,39)(34,40);(1,3,5,2)(4,7,8,6)(9,11,13,15)(10,14,12,16)(17,21,23,19)(18,20,24,22)(25,28)(26,33)(27,37)(29,31)(30,34)(32,39)(35,38)(36,40);(1,4,5,8)(2,6,3,7)(9,12,13,10)(11,16,15,14)(17,22,23,20)(18,19,24,21)(25,29,30,33)(26,34,31,28)(27,38,36,39)(32,40,35,37)

Polycyclic presentation:

2^9|a^2=gh,b^2=fhi,c^2=i,d^2=fg,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=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, 5658829);