#468940

order	512
nilpotency class	3
isoclinism	isoclinic class of #455576
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2730
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16635
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

726 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:

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

Polycyclic presentation:

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

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, 376136);