#395063

order	512
nilpotency class	3
isoclinism	isoclinic class of #394022
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1508 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)(9,10,16,11)(12,23,27,32)(13,18,29,22)(14,20,15,19)(17,26,21,25)(24,37,39,36)(28,34,30,33)(31,40,35,38);(2,5)(6,8)(9,11,16,10)(12,23,27,32)(13,22,29,18)(14,19,15,20)(17,26,21,25)(24,36,39,37)(28,33,30,34)(31,38,35,40);(9,12)(10,17)(11,21)(13,24)(14,26)(15,25)(16,27)(18,31)(19,32)(20,23)(22,35)(28,40)(29,39)(30,38)(33,36)(34,37);(1,3,4,7)(2,6,5,8)(9,13)(10,18)(11,22)(12,24)(14,30)(15,28)(16,29)(17,31)(19,34)(20,33)(21,35)(23,36)(25,40)(26,38)(27,39)(32,37)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,f^c=fi,g^c=gi,h^c=hi,f^d=fi

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, 302259);