#265167

order	512
nilpotency class	3
isoclinism	isoclinic class of #265163
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

895 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,d^2=e,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,d^b=dh,f^b=fhi,g^b=ghi,d^c=di,f^c=fi,g^c=ghi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 172363);