#381178

order	512
nilpotency class	3
isoclinism	isoclinic class of #30056
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2631
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14288
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1057 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,d^2=g,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,c^b=cg,f^b=fi,d^c=de,e^c=ei,g^c=gi,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, 288374);