#101380

order	512
nilpotency class	3
isoclinism	isoclinic class of #101343
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

626 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,c^2=h,d^2=gh,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,d^b=dh,e^b=ei,f^b=fh,d^c=di,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, 8576);