#120784

order	512
nilpotency class	3
isoclinism	isoclinic class of #10488
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#394
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

500 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 28 points:

(1,2,3,4)(5,6,10,7,8,12,9,11)(13,14,19,22)(15,23,26,27)(16,17,21,18)(20,25,28,24);(5,7,9,12,8,11,10,6)(13,15,18,25,19,26,17,24)(14,20,21,27,22,28,16,23);(6,11,12,7)(13,16,18,14,19,21,17,22)(15,20,25,27,26,28,24,23)

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=dh,d^2=hi,e^2=h,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=cf,d^b=di,f^b=fh,g^b=gi,f^c=fhi

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, 27980);