#268642

order	512
nilpotency class	3
isoclinism	isoclinic class of #268627
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

841 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=h,c^2=h,d^2=h,e^2=i,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,e^b=ei,f^b=fhi,g^b=gi,e^c=ehi,f^c=fhi,g^c=gi,e^d=ei,f^d=fhi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 175838);