#397349

order	512
nilpotency class	3
isoclinism	isoclinic class of #394382
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
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

1025 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=i,d^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,g^b=gi,f^c=fi,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, 304545);