#470947

order	512
nilpotency class	3
isoclinism	isoclinic class of #455962
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

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

678 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)(3,7)(4,5)(6,8)(9,10)(11,23)(12,29,31,38)(13,32,24,40)(14,21)(15,22)(16,20)(17,26)(18,30,35,25)(19,34,39,33)(27,36)(28,37);(2,5)(6,7)(9,11,14,26,15,28,16,27)(10,17,20,23,22,36,21,37)(12,13,25,34,31,24,30,33)(18,32,29,19,35,40,38,39);(1,3)(2,6)(4,8)(5,7)(9,12,15,31)(10,18)(11,24)(13,28)(14,30,16,25)(17,32,36,40)(19,23,39,37)(20,38)(21,29)(22,35)(26,34)(27,33);(3,8)(6,7)(9,13,15,24)(10,19,22,39)(11,25,28,30)(12,26,31,27)(14,34,16,33)(17,35,36,18)(20,40,21,32)(23,38,37,29)

Polycyclic presentation:

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

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, 378143);