#340945

order	512
nilpotency class	3
isoclinism	isoclinic class of #332071
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2541	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

628 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=e,c^2=h,d^2=ei,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=defi,f^a=fh,c^b=cg,d^b=de,g^b=gi,f^c=fh,g^c=gi,f^d=fh

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 248141);