#444919

order	512
nilpotency class	3
isoclinism	isoclinic class of #421447
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2474
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15794
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

627 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:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=ef,c^2=eg,d^2=e,e^2=i,f^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,f^c=fi,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, 352115);