#6444363

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438949
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	#35328	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2750 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=fi,c^2=fgi,d^2=f,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,h^b=hi,d^c=di

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, 6351559);