#394896

order	512
nilpotency class	3
isoclinism	isoclinic class of #394113
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

1348 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:

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

Polycyclic presentation:

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

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, 302092);