#470133

order	512
nilpotency class	3
isoclinism	isoclinic class of #454981
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2619
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16676
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

717 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=f,d^2=eg,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,f^b=fi,d^c=de,e^c=ei,h^c=hi,h^d=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 377329);