#342671

order	512
nilpotency class	3
isoclinism	isoclinic class of #334823
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	C₂×#547	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

692 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)(3,7)(4,6)(5,8)(9,10,14,15,13,17,11,18)(12,20,19,24,21,23,22,16)(25,26)(27,30)(28,31)(29,32)(33,40)(34,38)(35,37)(36,39);(1,3)(2,5)(4,8)(6,7)(9,11,13,14)(10,15,17,18)(12,19,21,22)(16,23,24,20)(25,27,28,34)(26,30,31,38)(29,33,36,35)(32,40,39,37);(9,12,13,21)(10,16,17,24)(11,19,14,22)(15,23,18,20)(26,31)(27,33)(29,36)(30,37)(34,35)(38,40);(2,6)(3,8)(10,15)(11,14)(16,23)(17,18)(19,22)(20,24)

Polycyclic presentation:

2^9|a^2=fi,b^2=hi,c^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, 249867);