#460569

order	512
nilpotency class	3
isoclinism	isoclinic class of #459756
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2761
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16367
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

912 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:

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

Polycyclic presentation:

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

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, 367765);