#466530

order	512
nilpotency class	3
isoclinism	isoclinic class of #454426
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

787 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=efg,c^2=ei,d^2=ef,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^c=de,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, 373726);