#201878

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

1612 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,g^2=hi,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, 109074);