#258854

order	512
nilpotency class	3
isoclinism	isoclinic class of #247527
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×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

730 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)(9,10,13,17)(11,14,18,22)(12,15,20,16)(19,21,24,23);(2,5)(6,7)(9,11,13,18)(10,14,17,22)(12,19,20,24)(15,21,16,23)(25,26,30,33)(27,34,37,40)(28,32,29,38)(31,35,39,36);(10,15)(12,20)(14,21)(16,17)(19,24)(22,23)(25,27,30,37)(26,31,33,39)(28,36,29,35)(32,40,38,34);(1,3,4,8)(2,6,5,7)(11,18)(14,22)(19,24)(21,23)(25,28,30,29)(26,32,33,38)(27,35,37,36)(31,34,39,40)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=i,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,g^b=gi,d^c=di,f^c=fh,g^c=gi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 166050);