#302184

order	512
nilpotency class	3
isoclinism	isoclinic class of #272667
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	C₂³×C₄	C₂⁴
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

846 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=ei,b^2=eh,c^2=f,d^2=eh,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=gh,g^c=ghi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 209380);