#480572

order	512
nilpotency class	4
isoclinism	isoclinic class of #480460
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×#41	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄ ◁ C₂×#520 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

922 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=hi,d^2=i,f^2=h,b^a=be,e^a=eg,g^a=gi,c^b=ch,e^b=ei,f^b=fg,d^c=di,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, 387768);