#405687

order	512
nilpotency class	3
isoclinism	isoclinic class of #394273
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1266 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=eh,d^2=e,b^a=be,c^a=cf,d^a=dh,c^b=cg,g^b=gi,h^c=hi,f^d=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 312883);