#5363293

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

1600 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fg,c^2=fh,d^2=f,e^2=fhi,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, 5270489);