#296492

order	512
nilpotency class	3
isoclinism	isoclinic class of #272667
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#2617	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

901 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)(9,10,14,16)(11,12,15,13)(17,18)(19,24)(20,30)(21,25)(22,27)(23,28)(26,31)(29,32);(2,5)(6,7)(9,11)(10,12)(13,16)(14,15)(17,19)(18,23)(20,31)(21,28)(22,29)(24,25)(26,32)(27,30);(9,11,12,16,14,15,13,10)(17,20,22,19,21,29,31,28)(18,24,26,30,25,23,27,32);(1,3,4,8)(2,6,5,7)(18,25)(23,24)(26,27)(30,32)

Polycyclic presentation:

2^9|a^2=eh,c^2=f,d^2=ehi,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, 203688);