#120581

order	512
nilpotency class	3
isoclinism	isoclinic class of #120577
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#391
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1441
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

549 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=d,d^2=h,e^2=hi,g^2=hi,b^a=bd,c^a=ce,d^a=dh,f^a=fh,c^b=cf,e^b=eh,e^c=ei,g^c=ghi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 27777);