#257799

order	512
nilpotency class	3
isoclinism	isoclinic class of #247527
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	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

715 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)(9,10,15,18)(11,16,20,24)(12,21,19,23)(13,17,14,22)(25,26)(27,30)(28,31)(29,33)(32,37)(34,39)(35,38)(36,40);(1,3,5,4)(2,6,7,8)(9,11)(10,16)(12,13)(14,19)(15,20)(17,21)(18,24)(22,23)(25,27,29,35)(26,30,33,38)(28,34,37,36)(31,39,32,40);(1,4,5,3)(2,6,7,8)(9,12)(10,16)(11,13)(14,20)(15,19)(17,23)(18,24)(21,22)(26,31)(27,34)(28,37)(32,33)(35,36)(39,40);(2,7)(6,8)(9,13,15,14)(10,17,18,22)(11,19,20,12)(16,23,24,21)(27,35)(30,38)(34,36)(39,40)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 164995);