#400791

order	512
nilpotency class	3
isoclinism	isoclinic class of #394044
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1310 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=ef,d^2=ef,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 307987);