#181063

order	512
nilpotency class	3
isoclinism	isoclinic class of #17392
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂²×#95
prime radical	C₂⁵	C₄²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1468 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)(3,9,10,6)(4,8)(5,7)(11,15,16,13)(12,14)(17,18,21,25,22,26,20,24)(19,23,27,30,29,31,28,32);(1,3,4,11,5,10,12,16)(2,6,8,13,7,9,14,15)(17,19)(18,23)(20,28)(21,27)(22,29)(24,32)(25,30)(26,31);(17,20,22,21)(18,24,26,25)(19,27,29,28)(23,30,31,32);(17,20,22,21)(18,25,26,24)(19,28,29,27)(23,30,31,32)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88259);