#304671

order	512
nilpotency class	3
isoclinism	isoclinic class of #271003
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=ehi,c^2=ef,d^2=ei,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=gi,d^c=di

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, 211867);