#6465634

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438943
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3155	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3234 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)(6,8)(9,10,14,21,16,22,15,20)(11,23,27,35,28,34,26,17)(12,18,13,19,30,37,29,36)(24,39,40,38,31,33,25,32);(2,4)(9,11)(10,17)(12,25)(13,31)(14,26)(15,27)(16,28)(18,33)(19,38)(20,23)(21,34)(22,35)(24,29)(30,40)(32,36)(37,39);(5,6,7,8)(9,12,15,29,16,30,14,13)(10,18,20,36,22,37,21,19)(11,24,26,25,28,31,27,40)(17,32,34,33,35,38,23,39);(9,13,14,30,16,29,15,12)(10,19,21,37,22,36,20,18)(11,25,27,24,28,40,26,31)(17,33,23,32,35,39,34,38);(9,14,16,15)(10,20,22,21)(11,26,28,27)(12,13,30,29)(17,23,35,34)(18,36,37,19)(24,25,31,40)(32,39,38,33)

Polycyclic presentation:

2^9|a^2=f,c^2=gh,d^2=hi,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

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, 6372830);