#6372956

order	512
nilpotency class	3
isoclinism	isoclinic class of #60454
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×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2706 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)(5,6,8,7)(9,10,14,18)(11,19,21,33)(12,16,23,30)(13,25,26,17)(15,27,29,22)(20,32,34,40)(24,37,36,31)(28,38,39,35);(9,11,13,22,14,21,26,27)(10,15,17,19,18,29,25,33)(12,20,24,35,23,34,36,38)(16,28,31,32,30,39,37,40);(2,4)(9,12)(10,16)(11,20)(13,24)(14,23)(15,28)(17,31)(18,30)(19,32)(21,34)(22,35)(25,37)(26,36)(27,38)(29,39)(33,40);(5,7,8,6)(11,21)(15,29)(19,33)(20,34)(22,27)(28,39)(32,40)(35,38);(12,23)(16,30)(20,34)(24,36)(28,39)(31,37)(32,40)(35,38)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343347, #72343359, #72343372, …
maximal normal subgroup of	#72343347, #72343359, #72343372, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6280152);