#6465631

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×#3147	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3148 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gh,d^2=hi,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, 6372827);