#256249

order	512
nilpotency class	3
isoclinism	isoclinic class of #246591
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	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

949 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,17)(13,16)(14,18)(19,22)(20,24)(21,23)(25,26,30,34)(27,35,37,39)(28,33,29,32)(31,36,40,38);(2,5)(6,8)(9,11,14,20)(10,15,18,24)(12,19,13,21)(16,23,17,22)(27,36)(28,29)(31,39)(32,33)(35,40)(37,38);(1,3,4,7)(2,6,5,8)(10,16)(12,13)(15,22)(17,18)(19,21)(23,24)(25,27)(26,31)(28,36)(29,38)(30,37)(32,39)(33,35)(34,40);(2,5)(6,8)(9,12,14,13)(10,17,18,16)(11,19,20,21)(15,22,24,23)(27,37)(31,40)(35,39)(36,38)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,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=gh,c^b=cg,f^b=fh,g^b=gi,d^c=dh,f^c=fh,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 163445);