#181274

order	512
nilpotency class	3
isoclinism	isoclinic class of #181131
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

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

1233 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=i,e^2=hi,f^2=h,g^2=hi,b^a=be,e^a=eh,g^a=ghi,c^b=ch,f^b=fi,d^c=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88470);