#202000

order	512
nilpotency class	3
isoclinism	isoclinic class of #201918
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1370 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)(3,6)(4,11,12,7)(5,8)(9,13,15,16)(10,14)(17,18,20,24,22,26,21,25)(19,27,29,23,28,31,30,32);(1,3,5,10)(2,6,8,14)(4,9,12,15)(7,13,11,16)(17,19)(18,23,26,32)(20,29)(21,30)(22,28)(24,31,25,27);(1,4)(2,7)(3,9)(5,12)(6,13)(8,11)(10,15)(14,16)(18,24)(20,21)(23,31)(25,26)(27,32)(29,30);(19,28)(23,32)(27,31)(29,30)

Polycyclic presentation:

2^9|a^2=g,b^2=egi,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fhi,d^b=dh,f^b=fi,e^c=ei,f^c=fh,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, 109196);