#111956

order	512
nilpotency class	3
isoclinism	isoclinic class of #111945
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

993 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72338210, #72338221, #72338222
maximal normal subgroup of	#72338210, #72338221, #72338222

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 19152);