#296490

order	512
nilpotency class	3
isoclinism	isoclinic class of #272667
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	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

977 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 203686);