#100498

order	512
nilpotency class	3
isoclinism	isoclinic class of #100235
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#386
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

393 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=gh,d^2=gh,e^2=i,f^2=hi,b^a=bd,c^a=ce,d^a=dg,d^b=dh,e^b=ei,f^b=fh,d^c=di,e^c=ei,f^c=fi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7694);