#121492

order	512
nilpotency class	3
isoclinism	isoclinic class of #9996
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#392
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

700 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257811, #72257884, #72257886, …
maximal normal subgroup of	#72257811, #72257884, #72257886, …
maximal quotient of	#72264829, #72265385

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 28688);