#304636

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

757 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)(11,19)(12,18)(13,15)(14,17)(16,22)(20,23)(21,24)(25,26)(27,33)(28,39)(29,36)(30,35)(31,34)(32,37)(38,40);(1,3,5,4)(2,6,8,7)(10,15)(16,23)(17,18)(19,24)(25,27)(26,32)(28,29)(30,37)(31,38)(33,35)(34,39)(36,40);(1,3,5,4)(2,7,8,6)(9,11,14,22,13,21,12,20)(10,16,18,19,15,23,17,24)(25,28,29,27,30,38,31,37)(26,33,34,39,35,32,36,40);(1,4,5,3)(2,7,8,6)(9,12,13,14)(10,17,15,18)(11,20,21,22)(16,24,23,19)(25,29,30,31)(26,34,35,36)(27,28,37,38)(32,39,33,40)

Polycyclic presentation:

2^9|a^2=eh,b^2=eh,c^2=efh,d^2=e,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, 211832);