#316578

order	512
nilpotency class	3
isoclinism	isoclinic class of #262883
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#24152	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

729 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)(3,7)(4,6)(5,8)(9,10)(11,15)(12,17)(13,22)(14,18)(16,19)(20,24)(21,23)(25,26)(27,31)(28,32)(29,34)(30,33)(35,40)(36,38)(37,39);(1,3,4,8)(2,5,6,7)(9,11)(10,15)(12,21)(13,19)(14,20)(16,22)(17,23)(18,24)(25,27,28,35,30,36,29,37)(26,31,32,40,33,38,34,39);(9,11)(10,16)(12,19)(13,21)(14,20)(15,22)(17,24)(18,23)(26,32)(27,35)(28,29)(31,38)(33,34)(36,37);(2,6)(5,7)(9,12,14,13)(10,17,18,22)(11,19,20,21)(15,16,24,23)(26,33)(27,36)(32,34)(35,37)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 223774);