#438871

order	512
nilpotency class	3
isoclinism	isoclinic class of #417778
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

807 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)(9,10,12,14)(11,15,16,13)(17,18)(19,20)(21,27)(22,28)(23,29)(24,25)(26,31)(30,32);(1,3)(2,6)(4,7)(5,8)(17,19,22,26,21,25,23,30)(18,24,28,32,27,20,29,31);(1,2,5,4)(3,6,8,7)(9,11,12,16)(10,13,14,15)(17,20)(18,25)(19,27)(21,24)(22,32)(23,31)(26,28)(29,30);(1,4,5,2)(3,6,8,7)(17,20)(18,26)(19,29)(21,24)(22,31)(23,32)(25,28)(27,30)

Polycyclic presentation:

2^9|a^2=ef,b^2=g,c^2=ef,d^2=ei,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=dei,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, 346067);