#437573

order	512
nilpotency class	3
isoclinism	isoclinic class of #419464
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2557
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15603
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

705 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)(3,15,19,9)(4,20,23,31)(5,24)(6,13)(7,14)(8,12)(10,22,30,17)(11,25)(16,29)(18,28,26,27)(21,32);(1,3,6,18,8,19,7,26)(2,9,13,27,12,15,14,28)(4,21,22,16,23,5,17,25)(10,29,20,24,30,11,31,32);(1,4,6,22,8,23,7,17)(2,10,14,31,12,30,13,20)(3,16,18,5,19,25,26,21)(9,24,28,29,15,32,27,11);(1,5,6,25,8,21,7,16)(2,11,13,32,12,29,14,24)(3,17,18,4,19,22,26,23)(9,20,27,30,15,31,28,10)

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=ef,d^2=f,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,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, 344769);