#457757

order	512
nilpotency class	3
isoclinism	isoclinic class of #457680
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2665
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16301
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

813 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:

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

Polycyclic presentation:

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

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, 364953);