#481016

order	512
nilpotency class	4
isoclinism	isoclinic class of #480872
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄⋊C₄	C₂⁴
prime radical	#2867	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄ ◁ C₂×#559 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

947 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)(19,23)(20,24,28,22)(21,25)(26,30)(27,31,32,29);(18,22,25,24)(23,29,30,31);(1,3,6,12)(2,7,10,15)(4,13,5,11)(8,16,9,14)(17,19)(18,23)(20,27)(21,26)(22,29)(24,31)(25,30)(28,32);(1,4,6,5)(2,8,10,9)(3,11,12,13)(7,14,15,16)(19,26)(23,30)(27,32)(29,31)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 388212);