#461953

order	512
nilpotency class	3
isoclinism	isoclinic class of #455397
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2700
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16410
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

588 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=ef,d^2=e,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,g^b=gi,d^c=de,g^c=gi,h^c=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, 369149);