#5805057

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

1776 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=fghi,c^2=ghi,d^2=fg,e^2=fgh,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, 5712253);