#480657

order	512
nilpotency class	4
isoclinism	isoclinic class of #480656
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂×C₂²⋊C₄	C₂⁴
prime radical	C₂⁶	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ C₂³×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1402 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387853);