#268065

order	512
nilpotency class	3
isoclinism	isoclinic class of #268043
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

916 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:

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

Polycyclic presentation:

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

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, 175261);