#420631

order	512
nilpotency class	3
isoclinism	isoclinic class of #417704
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1023 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=e,c^2=e,d^2=ei,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,g^b=gi,f^c=fi,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, 327827);