#1661799

order	512
nilpotency class	2
isoclinism	isoclinic class of #1506685
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

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

1665 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=fgh,d^2=g,e^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1568995);