#196344

order	512
nilpotency class	3
isoclinism	isoclinic class of #196344
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 ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1650 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)(4,15,18,27,16,28,31,30)(6,10,21,23,22,13,25,19)(9,20,26,32,12,24,29,17);(1,3,5,19,7,2,8,23)(4,12,20,29,16,9,24,26)(6,13,21,14,22,10,25,11)(15,31,27,17,28,18,30,32);(1,4)(2,9)(3,12)(5,20)(6,17)(7,16)(8,24)(10,15)(11,27)(13,28)(14,30)(18,21)(19,29)(22,32)(23,26)(25,31);(4,16)(9,12)(15,28)(17,32)(18,31)(20,24)(26,29)(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=fh,e^b=eh,f^b=fh,g^b=gh,d^c=di,e^c=eh,g^c=gh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 103540);