#6442596

order	512
nilpotency class	3
isoclinism	isoclinic class of #6442596
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#535
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂	C₂×#2408
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4439 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)(3,10)(4,17)(5,12)(6,11)(7,13)(8,15)(9,19)(14,24)(16,27)(18,28)(20,23)(21,26)(22,29)(25,31)(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|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	#72215784, #72215906, #72219688, …
maximal normal subgroup of	#72215784, #72215906, #72219688, …
maximal quotient of	#72214895, C₂×#6442596

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6349792);