#480560

order	512
nilpotency class	4
isoclinism	isoclinic class of #480454
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	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1037 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

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, 387756);