#468155

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	#16602
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

936 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)(3,12)(4,16)(5,7)(6,11)(8,15)(9,19)(10,21)(13,28)(14,22)(17,26)(18,27)(20,24)(23,32)(25,29)(30,31);(2,7,11,10)(8,24,12,22)(9,23,16,27)(25,30,28,26);(1,3)(2,8,11,12)(4,17)(5,20)(6,15)(7,22,10,24)(9,26,16,30)(13,18)(14,21)(19,31)(23,28,27,25)(29,32);(1,4,5,18,6,19,21,32)(2,9,10,27,11,16,7,23)(3,13,14,17,15,29,20,31)(8,25,22,26,12,28,24,30)

Polycyclic presentation:

2^9|b^2=g,c^2=g,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^c=de,e^c=ei,f^c=fi,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, 375351);