#180893

order	512
nilpotency class	2
isoclinism	isoclinic class of #180873
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂²×#95
prime radical	C₂²×#41	Q₈
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1289 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,d^2=i,e^2=h,f^2=i,g^2=h,b^a=be,g^a=gh,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, 88089);