#453887

order	512
nilpotency class	3
isoclinism	isoclinic class of #427387
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2760
commutator, Frattini, prime radical, cosocle	C₂⁵	C₂⁴
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

802 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:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=eh,c^2=efgi,d^2=ef,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,h^c=hi,e^d=ei,f^d=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 361083);