#427105

order	512
nilpotency class	3
isoclinism	isoclinic class of #418484
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

807 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 334301);