#6465654

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438997
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 ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3142 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)(9,10,14,13,16,20,15,19)(11,21,23,33,25,31,24,17)(12,18,29,27,28,35,30,36)(22,37,39,40,38,34,26,32);(2,4)(9,11,16,25)(10,17,20,33)(12,26,28,39)(13,31,19,21)(14,24,15,23)(18,34,35,37)(22,29,38,30)(27,40,36,32);(5,6,7,8)(9,12,15,30,16,28,14,29)(10,18,19,36,20,35,13,27)(11,22,24,26,25,38,23,39)(17,32,31,34,33,40,21,37);(9,13,14,20,16,19,15,10)(11,17,23,21,25,33,24,31)(12,27,29,35,28,36,30,18)(22,32,39,37,38,40,26,34);(12,28)(18,35)(22,38)(26,39)(27,36)(29,30)(32,40)(34,37)

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,d^a=di,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei

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, 6372850);