#6465640

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438949
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×#3154	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3164 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)(12,18,30,37,31,38,13,19)(24,33,25,39,40,36,29,32);(2,4)(9,11)(10,17)(12,29)(13,24)(14,26)(15,27)(16,28)(18,36)(19,32)(20,23)(21,34)(22,35)(25,31)(30,40)(33,38)(37,39);(5,6,7,8)(9,12,15,13,16,31,14,30)(10,18,20,19,22,38,21,37)(11,24,26,29,28,40,27,25)(17,32,34,36,35,39,23,33);(9,13,15,31,16,30,14,12)(10,19,20,38,22,37,21,18)(11,25,26,24,28,29,27,40)(17,33,34,32,35,36,23,39);(9,14,16,15)(10,20,22,21)(11,26,28,27)(12,30,31,13)(17,23,35,34)(18,19,38,37)(24,29,40,25)(32,33,39,36)

Polycyclic presentation:

2^9|a^2=f,c^2=gh,d^2=h,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, 6372836);