#406156

order	512
nilpotency class	3
isoclinism	isoclinic class of #395586
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2411
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14925
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

829 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 313352);