#316706

order	512
nilpotency class	3
isoclinism	isoclinic class of #262915
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

763 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)(11,20)(12,19)(13,18)(14,17)(15,22)(16,21)(23,24)(25,26)(27,35)(28,33)(29,32)(30,34)(31,37)(36,39)(38,40);(1,3,4,8)(2,5,6,7)(9,11)(10,15)(12,23)(13,22)(14,21)(16,19)(17,24)(18,20)(25,27,30,38,28,37,29,36)(26,31,34,39,33,35,32,40);(9,11)(10,16)(12,21)(13,22)(14,23)(15,19)(17,20)(18,24)(26,32)(27,36)(29,30)(31,35)(33,34)(37,38);(2,6)(5,7)(9,12,13,14)(10,17,18,19)(11,21,22,23)(15,16,20,24)

Polycyclic presentation:

2^9|b^2=egh,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=dhi,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, 223902);