#1361366

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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1901 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)(10,13)(11,15)(17,18,21,20)(19,22,23,24)(25,26)(27,30)(28,33)(29,34)(31,36)(32,38)(35,39)(37,40);(1,2,5,4)(3,6,7,8)(19,23)(22,24)(26,27)(31,33)(32,34)(39,40);(9,10)(11,14)(12,13)(15,16)(17,19)(18,22)(20,24)(21,23)(25,27,30,26)(28,33,36,31)(29,34,38,32)(35,40,37,39);(1,3)(2,6)(4,8)(5,7)(9,10)(11,14)(12,13)(15,16)(25,28)(26,31)(27,33)(29,37)(30,36)(32,40)(34,39)(35,38);(1,4,5,2)(3,8,7,6)(9,11,12,15)(10,14,13,16)(17,20,21,18)(19,24,23,22)(25,29)(26,32)(27,34)(28,35)(30,38)(31,39)(33,40)(36,37)

Polycyclic presentation:

2^9|a^2=fh,b^2=gi,c^2=f,e^2=hi,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, 1268562);