#343453

order	512
nilpotency class	3
isoclinism	isoclinic class of #333271
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

527 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)(3,4)(5,7)(6,8)(9,10,14,15,13,18,11,16)(12,20,19,17,22,23,21,24)(25,26)(27,30)(28,31)(29,32)(33,40)(34,38)(35,37)(36,39);(1,3)(2,6)(4,7)(5,8)(9,11,13,14)(10,15,18,16)(12,21,22,19)(17,23,24,20)(25,27)(26,30)(28,34)(29,33)(31,38)(32,40)(35,36)(37,39);(1,4,5,6)(2,3,7,8)(10,16)(11,14)(15,18)(17,23)(19,21)(20,24)(26,31)(27,33)(29,36)(30,37)(34,35)(38,40);(2,7)(3,8)(9,12,13,22)(10,17,18,24)(11,19,14,21)(15,20,16,23)(26,31)(30,38)(32,39)(37,40)

Polycyclic presentation:

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

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, 250649);