#451734

order	512
nilpotency class	3
isoclinism	isoclinic class of #420424
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2699
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16011
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

697 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)(3,6)(4,14,18,8)(5,19,21,9)(7,11)(10,22)(12,16,24,17)(13,20,25,31)(15,23)(26,28)(27,30)(29,32);(1,3,6,11,7,10,22,2)(4,14,16,24,18,8,17,12)(5,9,20,13,21,19,31,25)(15,26,28,27,30,32,29,23);(1,4,6,16,7,18,22,17)(2,8,3,12,11,14,10,24)(5,15,20,28,21,30,31,29)(9,23,13,26,19,27,25,32);(1,5,6,20,7,21,22,31)(2,9,10,25,11,19,3,13)(4,15,16,28,18,30,17,29)(8,23,24,32,14,27,12,26)

Polycyclic presentation:

2^9|a^2=efgh,b^2=ef,c^2=ef,d^2=e,e^2=i,g^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,e^b=ei,f^b=fi,g^b=gi,h^b=hi,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, 358930);