#324763

order	512
nilpotency class	3
isoclinism	isoclinic class of #279051
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

612 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)(9,10,14,19)(11,21,17,16)(12,23,24,22)(13,20,15,18)(25,26)(27,33)(28,31)(29,32)(30,35)(34,38)(36,40)(37,39);(2,5)(6,7)(9,11)(10,16)(12,20)(13,23)(14,17)(15,22)(18,24)(19,21)(25,27,29,36,28,35,37,34)(26,30,32,38,31,33,39,40);(9,12,13,16,14,24,15,21)(10,17,18,22,19,11,20,23)(26,31)(27,34)(29,37)(30,38)(33,40)(35,36);(1,3,4,8)(2,6,5,7)(9,13,14,15)(10,18,19,20)(11,22,17,23)(12,21,24,16)(26,31)(27,35)(32,39)(34,36)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 231959);