#303382

order	512
nilpotency class	3
isoclinism	isoclinic class of #273051
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

905 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=ef,d^2=ei,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=gh,d^c=dhi,g^c=ghi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 210578);