#469044

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2730
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16641
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

565 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:

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

Polycyclic presentation:

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

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, 376240);