#242046

order	512
nilpotency class	3
isoclinism	isoclinic class of #234044
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	#37626	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

889 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)(11,16)(12,21)(13,18)(14,23)(15,19)(17,22)(20,24)(25,26)(27,31)(28,33)(29,32)(30,34)(35,38)(36,40)(37,39);(2,5)(6,8)(9,11)(10,16)(12,15)(13,22)(14,20)(17,18)(19,21)(23,24)(25,27)(26,31)(28,36)(29,35)(30,37)(32,38)(33,40)(34,39);(9,12,15,11)(10,17,19,24)(13,22,14,20)(16,23,21,18)(26,32)(27,35)(28,29)(33,34)(36,37)(38,40);(1,3,4,7)(2,6,5,8)(9,13,15,14)(10,18,19,23)(11,20,12,22)(16,24,21,17)(25,28,30,29)(26,33,34,32)(27,35,37,36)(31,38,39,40)

Polycyclic presentation:

2^9|c^2=hi,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,g^a=gi,c^b=cg,d^b=dh,f^b=fi,g^b=gi,d^c=di,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, 149242);