#6437669

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437521
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4625 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)(5,6)(7,8)(9,10)(11,16)(12,17)(13,18)(14,20)(15,19)(21,31)(22,32)(23,34)(24,33)(25,35)(26,36)(27,37)(28,38)(29,40)(30,39);(2,4)(9,11,15,24)(10,16,19,33)(12,25,13,27)(14,29,30,23)(17,35,18,37)(20,40,39,34)(21,28,22,26)(31,38,32,36);(6,8)(9,12,15,13)(10,17,19,18)(11,21,24,22)(14,28,30,26)(16,31,33,32)(20,38,39,36)(23,27,29,25)(34,37,40,35);(9,13,15,12)(10,18,19,17)(11,22,24,21)(14,26,30,28)(16,32,33,31)(20,36,39,38)(23,25,29,27)(34,35,40,37);(10,19)(16,33)(17,18)(20,39)(31,32)(34,40)(35,37)(36,38)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6344865);