#452774

order	512
nilpotency class	3
isoclinism	isoclinic class of #424610
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2477
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16063
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

635 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:

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

Polycyclic presentation:

2^9|a^2=egh,b^2=fg,c^2=ef,d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,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, 359970);