#304860

order	512
nilpotency class	3
isoclinism	isoclinic class of #271771
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

700 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)(10,14)(11,18)(12,19)(15,22)(20,24)(21,23)(25,26,29,32)(27,34,35,31)(28,36,38,40)(30,39,37,33);(1,3,5,4)(2,6,8,7)(9,10)(11,15)(12,17)(13,14)(16,19)(18,23)(20,21)(22,24)(25,27)(26,31)(28,37)(29,35)(30,38)(32,34)(33,40)(36,39);(1,3,5,4)(2,7,8,6)(9,11,19,24,13,20,12,18)(10,15,16,22,14,21,17,23)(25,28,37,27,29,38,30,35)(26,31,39,40,32,34,33,36);(1,4,5,3)(2,7,8,6)(9,12,13,19)(10,16,14,17)(11,18,20,24)(15,22,21,23)(26,32)(27,35)(28,38)(33,39)

Polycyclic presentation:

2^9|a^2=ehi,b^2=eh,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,d^b=dh,f^b=fi,g^b=gi,d^c=di

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 212056);