#121532

order	512
nilpotency class	3
isoclinism	isoclinic class of #9914
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#120
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	#385
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

608 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 28728);