#6372051

order	512
nilpotency class	3
isoclinism	isoclinic class of #60367
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹×C₄

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	C₂³×#41	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

5094 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72316163, #72316187
maximal normal subgroup of	#72316163, #72316187

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6279247);