#197181

order	512
nilpotency class	3
isoclinism	isoclinic class of #197020
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	#60802	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1099 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egi,d^2=h,e^2=hi,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,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

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 104377);