#200314

order	512
nilpotency class	3
isoclinism	isoclinic class of #200158
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	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#121	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1064 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egi,d^2=hi,e^2=hi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fhi,d^b=dh,e^b=ei,f^b=fh,g^b=gi,d^c=dh,e^c=eh,f^c=fi,g^c=gh

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, 107510);