#102869

order	512
nilpotency class	3
isoclinism	isoclinic class of #102857
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#127
commutator, Frattini, cosocle	C₄³	C₂³
socle, prime radical	C₂³	#392
LCS	G ▷ C₄³ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

473 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=g,d^2=g,e^2=i,f^2=hi,b^a=bd,c^a=ce,d^a=dg,e^a=ei,f^a=fgi,c^b=cf,d^b=dg,e^b=eghi,f^b=fhi,d^c=dh,e^c=ei,f^c=fhi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72211235, C₂×#102869, #72211663
maximal normal subgroup of	#72211235, C₂×#102869, #72211663
maximal quotient of	#72211228, C₂×#102869

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10065);