#395057

order	512
nilpotency class	3
isoclinism	isoclinic class of #393975
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
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

1552 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,22)(13,25)(14,18)(16,23)(17,28)(19,36)(20,34)(21,31)(24,35)(26,30)(27,33)(29,37)(32,39)(38,40);(2,5)(6,8)(9,11,14,21)(10,15,18,31)(12,19,24,38)(13,26,28,20)(16,32,33,29)(17,34,25,30)(22,36,35,40)(23,39,27,37);(9,12)(10,16)(11,19)(13,27)(14,24)(15,29)(17,35)(18,33)(20,39)(21,38)(22,25)(23,28)(26,37)(30,40)(31,32)(34,36);(1,3,4,7)(2,6,5,8)(13,28)(17,25)(20,26)(22,35)(23,27)(30,34)(36,40)(37,39)

Polycyclic presentation:

2^9|b^2=i,d^2=e,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,f^b=fi,f^c=fi,h^c=hi,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, 302253);