#345271

order	512
nilpotency class	3
isoclinism	isoclinic class of #334823
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

588 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 252467);