#422612

order	512
nilpotency class	3
isoclinism	isoclinic class of #422611
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15308
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1007 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 329808);