#316855

order	512
nilpotency class	3
isoclinism	isoclinic class of #262851
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

582 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 224051);