#481031

order	512
nilpotency class	4
isoclinism	isoclinic class of #480860
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	#36517	C₂
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄ ◁ C₂×#553 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

933 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)(20,27,28,22)(21,24)(25,31,32,29)(26,30);(18,22,24,27)(20,28)(23,29,30,31)(25,32);(1,3,6,12)(2,7,10,15)(4,13,5,11)(8,16,9,14)(17,19,21,26)(18,23,24,30)(20,25,28,32)(22,29,27,31);(1,4,6,5)(2,8,10,9)(3,11,12,13)(7,14,15,16)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=hi,d^2=h,f^2=h,b^a=be,e^a=eg,g^a=gi,e^b=ei,f^b=fg,d^c=dh,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, 388227);