#252914

order	512
nilpotency class	3
isoclinism	isoclinic class of #247527
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2856	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

995 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|b^2=eh,c^2=i,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,d^b=dh,f^b=fi,g^b=gi,f^c=fh,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, 160110);