#451445

order	512
nilpotency class	3
isoclinism	isoclinic class of #418871
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2697
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15998
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

776 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ef,c^2=ef,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,h^b=hi,e^c=ei,f^c=fi,g^d=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, 358641);