#404590

order	512
nilpotency class	3
isoclinism	isoclinic class of #394289
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

827 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=e,d^2=g,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,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, 311786);