#6465660

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438955
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×#3035	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3128 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)(9,10,14,20,15,21,16,22)(11,23,26,35,27,36,28,17)(12,29,32,37,31,18,13,19)(24,34,25,33,40,39,30,38);(2,4)(9,11,15,27)(10,17,21,35)(12,30,31,25)(13,24,32,40)(14,28,16,26)(18,34,29,39)(19,38,37,33)(20,36,22,23);(5,6,7,8)(9,12,14,32,15,31,16,13)(10,18,20,19,21,29,22,37)(11,24,26,25,27,40,28,30)(17,33,23,39,35,38,36,34);(9,13,16,31,15,32,14,12)(10,19,22,18,21,37,20,29)(11,25,28,24,27,30,26,40)(17,34,36,38,35,39,23,33);(12,31)(13,32)(18,29)(19,37)(24,40)(25,30)(33,38)(34,39)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6372856);