#243597

order	512
nilpotency class	3
isoclinism	isoclinic class of #233019
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

595 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 150793);