#437606

order	512
nilpotency class	3
isoclinism	isoclinic class of #420065
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2686
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15605
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

694 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=ef,c^2=ef,d^2=f,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,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, 344802);