#116172

order	512
nilpotency class	3
isoclinism	isoclinic class of #116168
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

655 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,e^2=h,f^2=i,g^2=hi,b^a=bd,c^a=ce,e^a=ei,c^b=cf,e^b=eh,f^b=fhi,d^c=dh,e^c=eh,f^c=fi,g^c=ghi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239017
maximal normal subgroup of	#72239017

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 23368);