#480142

order	512
nilpotency class	4
isoclinism	isoclinic class of #33348
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂²×#99
commutator	C₂³	C₂³×C₈
Frattini, cosocle	C₂×M₁₆	C₂⁴
socle	C₂	C₂×#1459
prime radical	C₂×#525	C₄
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ C₂×#568 ◁ G
LECS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1000 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=f,c^2=h,f^2=h,h^2=i,b^a=be,e^a=eg,g^a=gi,f^b=fg,d^c=di,f^e=fi

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, 387338);