#1735684

order	512
nilpotency class	2
isoclinism	isoclinic class of #1506685
automorphism	?
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

1828 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)(3,7)(9,10,12,14)(11,13,15,16)(17,18,21,24)(19,23,20,22)(25,26)(27,33)(28,32)(29,30)(31,34)(35,40)(36,38)(37,39);(2,5)(6,8)(11,15)(13,16)(17,18,21,24)(19,22,20,23)(26,29)(27,34)(28,36)(39,40);(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,19,21,20)(18,22,24,23)(25,27)(26,31)(28,37)(29,33)(30,34)(32,40)(35,36)(38,39);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,15)(14,16)(17,18)(19,22)(20,23)(21,24)(25,28,30,36)(26,32,29,38)(27,35,34,37)(31,39,33,40);(1,3)(2,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,20,21,19)(18,23,24,22)(25,29,30,26)(27,33,34,31)(28,38,36,32)(35,40,37,39)

Polycyclic presentation:

2^9|a^2=gh,b^2=fhi,c^2=fhi,d^2=f,e^2=hi,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, 1642880);