#115092

order	512
nilpotency class	3
isoclinism	isoclinic class of #10046
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	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	#386
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

1028 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)(15,22,24,18)(16,21)(17,20)(19,23,27,25)(26,28);(2,4)(5,6,9,12)(7,11,8,10)(13,15,17,24)(14,18,20,22)(16,25,26,23)(19,21,27,28);(6,10)(7,8)(11,12)(13,15)(14,19)(16,25)(17,24)(18,21)(20,27)(22,28)(23,26)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72215258, #72216051, #72216054
maximal normal subgroup of	#72215258, #72216051, #72216054

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 22288);