#470946

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

Polycyclic presentation:

2^9|a^2=efg,b^2=efh,c^2=ef,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, 378142);