#6436631

order	512
nilpotency class	3
isoclinism	isoclinic class of #29634
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator, Frattini, cosocle	C₂⁴	C₂⁵
socle	C₂²	C₂²×#120
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3509 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72251116, #72251177, #72268363
maximal normal subgroup of	#72251116, #72251177, #72268363

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6343827);