#1851737

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

1501 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)(4,7,8,6)(9,10,13,12)(11,15,16,14)(25,26)(27,31)(28,34)(29,37)(30,33)(32,38)(35,40)(36,39);(2,3)(4,8)(18,22)(23,24)(25,27,30,28)(26,31,33,34)(29,35,38,36)(32,39,37,40);(2,3)(6,7)(9,10,13,12)(11,14,16,15)(17,18,21,22)(19,23,20,24)(25,28,30,27)(26,31,33,34)(29,36,38,35)(32,39,37,40);(1,3,5,2)(4,7,8,6)(9,11,13,16)(10,14,12,15)(17,19,21,20)(18,23,22,24)(25,26)(27,34)(28,31)(29,32)(30,33)(35,40)(36,39)(37,38);(1,4)(2,6)(3,7)(5,8)(9,12,13,10)(11,15,16,14)(17,20,21,19)(18,24,22,23)(25,29)(26,32)(27,35)(28,36)(30,38)(31,39)(33,37)(34,40)

Polycyclic presentation:

2^9|a^2=fi,b^2=fg,c^2=fghi,d^2=fhi,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, 1758933);