#440962

order	512
nilpotency class	3
isoclinism	isoclinic class of #419459
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, prime radical, cosocle	C₂⁵	C₂⁴
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1202 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 348158);