#452646

order	512
nilpotency class	3
isoclinism	isoclinic class of #424610
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2477
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16057
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

775 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=g,c^2=ef,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,h^b=hi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,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, 359842);