#102901

order	512
nilpotency class	3
isoclinism	isoclinic class of #102889
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#392
commutator, Frattini, cosocle	C₄³	C₂³
LCS	G ▷ C₄³ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

486 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=h,d^2=g,e^2=h,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,c^b=cf,d^b=dg,f^b=fi,e^c=eh,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10097);