#1338190

order	512
nilpotency class	2
isoclinism	isoclinic class of #1258513
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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1590 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)(9,10,14,12)(11,13,16,15)(25,26)(27,33)(28,34)(29,37)(30,31)(32,39)(35,36)(38,40);(9,10,14,12)(11,15,16,13)(17,18,20,22)(19,21,23,24)(26,31)(27,34)(32,37)(35,40);(9,11,14,16)(10,15,12,13)(17,18)(19,21)(20,22)(23,24)(25,27)(26,28)(29,35)(30,34)(31,33)(32,36)(37,38)(39,40);(1,2)(3,6)(4,5)(7,8)(9,12,14,10)(11,13,16,15)(25,28)(26,27)(29,38)(30,33)(31,34)(32,40)(35,37)(36,39);(1,3,4,7)(2,6,5,8)(9,13)(10,11)(12,16)(14,15)(17,19,20,23)(18,21,22,24)(25,29,30,39)(26,32,31,37)(27,35,34,40)(28,36,33,38)

Polycyclic presentation:

2^9|a^2=fg,b^2=gh,c^2=fg,d^2=fg,e^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=ef

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1245386);