#450924

order	512
nilpotency class	3
isoclinism	isoclinic class of #417538
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1298 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:

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

Polycyclic presentation:

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

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, 358120);