#201824

order	512
nilpotency class	3
isoclinism	isoclinic class of #201678
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#35050	C₂
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1622 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,e^b=eh,f^b=fi,g^b=gh,d^c=dh,e^c=ei,g^c=gi

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, 109020);