#474238

order	512
nilpotency class	3
isoclinism	isoclinic class of #455699
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2797
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16917
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

641 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=ef,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,e^b=ei,f^b=fi,g^b=gi,d^c=de,e^c=ei,g^d=gi,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, 381434);