#112884

order	512
nilpotency class	3
isoclinism	isoclinic class of #112884
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	#34609	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

1003 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)(3,9,11,15)(4,7)(5,8)(6,10,14,12)(13,16)(17,18)(19,27,30,32)(20,23)(21,25)(22,26)(24,29,31,28);(1,3)(2,6)(4,12)(5,11)(7,15)(8,14)(9,16)(10,13)(18,23)(20,21)(24,27)(25,26)(28,29)(31,32);(6,14)(9,15)(17,19)(18,24)(20,29)(21,28)(22,30)(23,27)(25,32)(26,31)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281679, #72293048, #72316788
maximal normal subgroup of	#72281679, #72293048, #72316788

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 20080);