#400000

order	512
nilpotency class	3
isoclinism	isoclinic class of #394055
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1418 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

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, 307196);