#1329422

order	512
nilpotency class	2
isoclinism	isoclinic class of #1258513
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

1526 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)(9,10,14,16)(11,12,15,13)(17,18,21,23)(19,20,22,24)(25,26)(27,32)(28,33)(29,34)(30,31)(35,37)(36,40)(38,39);(9,11,14,15)(10,12,16,13)(17,19)(18,20)(21,22)(23,24)(26,31)(27,33)(34,39)(35,40);(9,12,14,13)(10,11,16,15)(17,20)(18,22)(19,23)(21,24)(25,27,30,33)(26,32,31,28)(29,35,38,40)(34,37,39,36);(1,2)(3,6)(4,5)(7,8)(9,13,14,12)(10,15,16,11)(25,28,30,32)(26,33,31,27)(29,37,38,36)(34,35,39,40);(1,3,4,7)(2,6,5,8)(9,10)(11,12)(13,15)(14,16)(25,29,30,38)(26,34,31,39)(27,35,33,40)(28,36,32,37)

Polycyclic presentation:

2^9|a^2=fgh,b^2=g,c^2=fg,d^2=fg,e^2=fgi,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, 1236618);