#197092

order	512
nilpotency class	3
isoclinism	isoclinic class of #197026
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

1033 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=h,e^2=hi,f^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,d^b=dh,e^b=ei,f^b=fh,g^b=gi,d^c=di,e^c=eh,f^c=fh,g^c=gh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 104288);