#417355

order	512
nilpotency class	3
isoclinism	isoclinic class of #397902
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2397
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

997 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)(3,5)(4,6)(7,8)(9,10)(11,22)(12,28)(13,19)(14,35)(15,20)(16,21)(17,25)(18,30)(23,32)(24,31)(26,37)(27,36)(29,38)(33,39)(34,40);(1,3,4,7)(2,5,6,8)(9,11,14,25,16,27,15,26)(10,17,20,22,21,37,35,36)(12,19,29,39,31,40,30,23)(13,24,32,38,34,28,33,18);(3,7)(5,8)(9,12,14,30,16,31,15,29)(10,18,20,28,21,38,35,24)(11,23,26,19,27,39,25,40)(13,17,32,36,34,37,33,22);(2,6)(5,8)(9,13,15,33,16,34,14,32)(10,19,20,39,21,40,35,23)(11,24,26,18,27,28,25,38)(12,17,29,22,31,37,30,36)

Polycyclic presentation:

2^9|b^2=eh,c^2=f,d^2=gh,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,f^b=fi,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 324551);