#118337

order	512
nilpotency class	3
isoclinism	isoclinic class of #118335
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁶	Q₈
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

1040 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fi,c^b=cf,f^b=fi,d^c=di,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 25533);