#263753

order	512
nilpotency class	3
isoclinism	isoclinic class of #263739
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

843 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 170949);