#480670

order	512
nilpotency class	4
isoclinism	isoclinic class of #480670
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	C₂²×#120	C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×D₄ ◁ C₂×#518 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1299 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387866);