#6458273

order	512
nilpotency class	3
isoclinism	isoclinic class of #6442596
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#535
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂	C₂×#2487
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1771 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=g,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,h^a=hi,c^b=ch,d^b=dh,f^b=fi,e^c=ei,f^c=fi,f^d=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72338511, #72338519, #72338529
maximal normal subgroup of	#72338511, #72338519, #72338529

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6365469);