#469862

order	512
nilpotency class	3
isoclinism	isoclinic class of #460970
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2759
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16665
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

769 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)(3,12,15,28)(4,17)(5,10)(6,22)(7,11)(8,16,24,13)(9,20)(14,29,18,32)(19,27)(21,26)(23,31,25,30);(3,13,15,16)(8,12,24,28)(14,30,18,31)(23,29,25,32);(1,3,5,13,7,15,6,16)(2,8,10,28,11,24,22,12)(4,18,19,30,20,14,21,31)(9,25,26,32,17,23,27,29);(1,4,6,19,7,20,5,21)(2,9,10,27,11,17,22,26)(3,14,16,31,15,18,13,30)(8,23,28,32,24,25,12,29)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 377058);