#115767

order	512
nilpotency class	3
isoclinism	isoclinic class of #9914
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

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	#386
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

882 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)(7,9)(8,11)(10,12)(13,14,16,20,17,19,23,26)(15,18,22,25,21,24,27,28);(1,2,3,4)(5,6)(7,9)(8,11)(10,12)(13,15)(14,18)(16,22)(17,21)(19,24)(20,25)(23,27)(26,28);(6,9)(7,12)(10,11)(18,24)(25,28)

Polycyclic presentation:

2^9|a^2=g,b^2=d,e^2=h,f^2=h,g^2=hi,b^a=bd,c^a=ce,e^a=eh,f^a=fi,c^b=cf,e^b=ei,f^b=fh,e^c=eh,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, 22963);