#263170

order	512
nilpotency class	3
isoclinism	isoclinic class of #263147
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
prime radical	#18888	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

799 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:

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

Polycyclic presentation:

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

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, 170366);