#194360

order	512
nilpotency class	3
isoclinism	isoclinic class of #18866
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#386
prime radical	C₂⁴	#121
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1083 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=h,d^2=h,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dhi,d^b=dh,e^b=eh,f^b=fi,g^b=gh,d^c=dh,e^c=ei,f^c=fi,g^c=gi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 101556);