#470880

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

Polycyclic presentation:

2^9|a^2=g,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, 378076);