#474043

order	512
nilpotency class	3
isoclinism	isoclinic class of #453995
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2812
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16894
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

747 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:

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

Polycyclic presentation:

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

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, 381239);