#380596

order	512
nilpotency class	3
isoclinism	isoclinic class of #380594
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14271
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60454	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1466 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72338483, #72338488, #72338492
maximal normal subgroup of	#72338483, #72338488, #72338492

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 287792);