#243192

order	512
nilpotency class	3
isoclinism	isoclinic class of #233152
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

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

624 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,8)(4,6)(5,7)(9,10,13,18,14,19,12,17)(11,15,20,23,21,24,22,16)(25,26,28,33,30,34,29,32)(27,31,37,39,36,38,35,40);(1,3,5,4)(2,6,7,8)(9,11)(10,15)(12,22)(13,20)(14,21)(16,17)(18,23)(19,24)(25,27)(26,31)(28,37)(29,35)(30,36)(32,40)(33,39)(34,38);(1,4,5,3)(2,6,7,8)(9,11)(10,16)(12,20)(13,22)(14,21)(15,17)(18,24)(19,23)(26,32)(27,35)(28,29)(31,38)(33,34)(36,37);(9,12,14,13)(10,17,19,18)(11,20,21,22)(15,23,24,16)(27,36)(31,38)(35,37)(39,40)

Polycyclic presentation:

2^9|a^2=f,b^2=e,c^2=e,d^2=hi,f^2=h,g^2=i,b^a=be,c^a=cf,c^b=cg,d^b=dh,g^b=gi,f^c=fh,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 150388);