#307525

order	512
nilpotency class	3
isoclinism	isoclinic class of #264203
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	C₂²×D₈	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

613 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)(4,5)(6,8)(9,10)(11,16)(12,21)(13,23)(14,18)(15,19)(17,22)(20,24)(25,26,30,34)(27,31,37,40)(28,32,29,33)(35,38,36,39);(2,5)(6,7)(9,11,13,19,14,20,22,12)(10,15,17,16,18,21,23,24)(25,27)(26,31)(28,36)(29,35)(30,37)(32,39)(33,38)(34,40);(9,11,13,19,14,20,22,12)(10,15,17,16,18,21,23,24)(27,35)(28,29)(31,38)(32,33)(36,37)(39,40);(1,3,4,8)(2,6,5,7)(9,12,22,20,14,19,13,11)(10,16,23,15,18,24,17,21)(25,28,30,29)(26,32,34,33)(27,36,37,35)(31,39,40,38)

Polycyclic presentation:

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

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, 214721);