#384298

order	512
nilpotency class	3
isoclinism	isoclinic class of #384297
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14376
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

999 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284899, #72286242, #72286267
maximal normal subgroup of	#72284899, #72286242, #72286267

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 291494);