#452727

order	512
nilpotency class	3
isoclinism	isoclinic class of #452727
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16061
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

914 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:

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

Polycyclic presentation:

2^9|a^2=egh,b^2=efg,c^2=ef,f^2=i,g^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,g^b=gi,e^c=ei,f^c=fi,g^d=gi

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, 359923);