#461654

order	512
nilpotency class	3
isoclinism	isoclinic class of #454981
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2619
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16398
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

927 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:

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

Polycyclic presentation:

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

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, 368850);