#470881

order	512
nilpotency class	3
isoclinism	isoclinic class of #455641
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2826
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

699 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)(9,10,16,20,15,22,14,21)(11,23,26,17,28,40,27,34)(12,29,31,37,25,36,32,18)(13,19,30,38,24,39,33,35);(2,5)(6,8)(9,11,16,26,15,28,14,27)(10,17,20,40,22,34,21,23)(12,30,32,13,25,33,31,24)(18,35,36,39,37,38,29,19);(1,3)(2,6)(4,7)(5,8)(9,12,14,32,15,25,16,31)(10,18,20,29,22,37,21,36)(11,24,26,33,28,13,27,30)(17,19,23,35,34,39,40,38);(3,7)(6,8)(9,13)(10,19)(11,25)(12,28)(14,30)(15,24)(16,33)(17,18)(20,35)(21,38)(22,39)(23,29)(26,31)(27,32)(34,37)(36,40)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 378077);