#295279

order	512
nilpotency class	3
isoclinism	isoclinic class of #272667
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	C₂³×C₄	C₂⁴
prime radical	#39738	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1038 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)(17,18,21,24)(19,25,27,22)(20,23,29,31)(26,30,28,32);(2,5)(6,7)(9,10)(11,13)(12,15)(14,16)(17,19)(18,22)(20,28)(21,27)(23,30)(24,25)(26,29)(31,32);(10,13)(11,16)(14,15)(17,20)(18,23)(19,26,27,28)(21,29)(22,30,25,32)(24,31);(1,3,4,8)(2,6,5,7)(18,24)(22,25)(23,31)(30,32)

Polycyclic presentation:

2^9|a^2=e,c^2=f,d^2=eh,g^2=i,b^a=be,c^a=cf,d^a=de,g^a=gh,c^b=cg,f^b=fh,g^b=gi,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, 202475);