#181092

order	512
nilpotency class	3
isoclinism	isoclinic class of #17411
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂²×#95
prime radical	C₂⁴	C₂³×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1326 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,d^2=h,f^2=i,g^2=i,b^a=be,e^a=eh,f^b=fh,d^c=dhi

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88288);