#304572

order	512
nilpotency class	3
isoclinism	isoclinic class of #270875
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

796 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=eh,b^2=ehi,c^2=efh,d^2=ehi,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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 211768);