#110548

order	512
nilpotency class	3
isoclinism	isoclinic class of #110532
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1437
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

671 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 32 points:

(1,2)(3,7,11,15)(4,6)(5,9)(8,12)(10,13,16,14)(17,18,22,25,23,26,21,20)(19,27,28,24,29,32,30,31);(2,6,9,8)(7,13,15,14)(17,19)(18,24)(20,27)(21,30)(22,28)(23,29)(25,32)(26,31);(1,3)(2,7,9,15)(4,10)(5,11)(6,13,8,14)(12,16)(17,20,22,26,23,25,21,18)(19,27,28,31,29,32,30,24)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 17744);