#6382779

order	512
nilpotency class	3
isoclinism	isoclinic class of #61049
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2740 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fh,c^2=g,d^2=fi,f^2=i,h^2=i,b^a=bf,c^a=cg,d^b=df,f^b=fi,h^b=hi,e^c=ei

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6289975);