#378906

order	512
nilpotency class	3
isoclinism	isoclinic class of #28182
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2535
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14225
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

895 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 286102);