#6454876

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437995
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

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 ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3324 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)(11,19)(12,16)(13,24)(14,18)(15,21)(17,20)(22,23);(9,10)(11,19,21,15)(12,22,23,16)(13,17)(14,18)(20,24);(2,5)(6,8)(9,11,14,21)(10,15,18,19)(12,20,23,13)(16,24,22,17);(9,12)(10,16)(11,20)(13,21)(14,23)(15,24)(17,19)(18,22);(1,3,4,7)(2,6,5,8)(9,13,14,20)(10,17,18,24)(11,12,21,23)(15,16,19,22)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6362072);