#437389

order	512
nilpotency class	3
isoclinism	isoclinic class of #419464
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2557
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15595
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

582 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=ef,d^2=f,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,g^b=gi,h^b=hi,g^c=gi,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, 344585);