#6448644

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437621
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

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

3713 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 24 points:

(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,15)(13,16)(17,20)(18,21)(19,23)(22,24);(9,11)(10,14)(12,17)(13,18)(15,22)(16,21)(19,24)(20,23);(2,5)(6,8)(9,12)(10,15)(11,17)(13,19)(14,20)(16,23)(18,24)(21,22);(1,3,4,7)(2,6,5,8)(9,12)(10,15)(11,17)(13,19)(14,20)(16,23)(18,24)(21,22);(11,18)(14,21)(17,24)(20,22)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257469, #72257485, #72257536
maximal normal subgroup of	#72257469, #72257485, #72257536
maximal quotient of	#72264356

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6355840);