#255277

order	512
nilpotency class	3
isoclinism	isoclinic class of #246239
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×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

896 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)(3,6)(4,5)(7,8)(9,10)(11,12)(13,17)(14,18)(15,19)(16,23)(20,22)(21,24)(25,26)(27,30)(28,31)(29,32)(33,37)(34,36)(35,38)(39,40);(2,5)(6,8)(9,11)(10,12)(13,20)(14,16)(15,21)(17,22)(18,23)(19,24)(25,27,29,34)(26,30,32,36)(28,33,35,39)(31,37,38,40);(1,3,4,7)(2,6,5,8)(9,12)(10,16)(11,17)(13,22)(14,23)(15,24)(18,21)(19,20)(28,35)(30,36)(31,38)(33,39);(2,5)(6,8)(9,13,15,14)(10,17,19,18)(11,20,21,16)(12,22,24,23)(25,28,29,35)(26,31,32,38)(27,33,34,39)(30,37,36,40)

Polycyclic presentation:

2^9|b^2=i,c^2=e,d^2=hi,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=di,f^c=fh,g^c=gh

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 162473);