#465052

order	512
nilpotency class	3
isoclinism	isoclinic class of #458124
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2731
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16514
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

933 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)(3,13)(4,17)(5,8)(6,11)(7,12)(9,15)(10,21)(14,27)(16,24)(18,25)(19,28)(20,23)(22,26)(29,30)(31,32);(2,8,12,11)(9,24,26,13)(10,23,17,28)(25,30,32,27);(1,3,6,15,7,16,5,22)(2,9,8,13,12,26,11,24)(4,18,20,29,21,31,19,14)(10,27,23,32,17,30,28,25);(1,4,5,19,7,21,6,20)(2,10,11,28,12,17,8,23)(3,14,15,18,16,29,22,31)(9,25,13,27,26,32,24,30)

Polycyclic presentation:

2^9|b^2=g,c^2=f,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,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, 372248);