#442014

order	512
nilpotency class	3
isoclinism	isoclinic class of #418500
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

804 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

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, 349210);