#291010

order	512
nilpotency class	3
isoclinism	isoclinic class of #268355
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

911 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)(17,18,23,27)(19,24,25,31)(20,29,32,28)(21,26,22,30);(1,3,4,8)(2,5,6,7)(10,13)(11,15)(14,16)(17,19,23,25)(18,24,27,31)(20,30,32,26)(21,28,22,29);(9,10,12,14)(11,16,15,13)(17,20,21,31,23,32,22,24)(18,25,30,28,27,19,26,29);(2,6)(5,7)(17,21,23,22)(18,26,27,30)(19,28,25,29)(20,31,32,24)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 198206);