#399089

order	512
nilpotency class	3
isoclinism	isoclinic class of #393925
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

898 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 306285);