#6439000

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438997
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#35659	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3184 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=f,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dhi,e^c=ei,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6346196);