#435543

order	512
nilpotency class	3
isoclinism	isoclinic class of #418818
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2684
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15529
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

749 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)(3,13)(4,18,21,32)(5,10)(6,26,28,12)(8,17)(9,27,30,22)(14,23)(15,20)(16,29)(19,25)(24,31);(1,3,6,15,7,17,28,16)(2,8,26,29,11,13,12,20)(4,19,27,31,21,10,22,14)(5,9,23,18,25,30,24,32);(1,4,6,27,7,21,28,22)(2,9,26,18,11,30,12,32)(3,14,15,19,17,31,16,10)(5,20,23,8,25,29,24,13);(1,5)(2,10)(3,9)(4,20)(6,23)(7,25)(8,27)(11,19)(12,31)(13,22)(14,26)(15,18)(16,32)(17,30)(21,29)(24,28)

Polycyclic presentation:

2^9|a^2=fg,b^2=ef,c^2=ef,f^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,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,g^d=gi

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, 342739);