#403399

order	512
nilpotency class	3
isoclinism	isoclinic class of #394046
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2237 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ei,b^2=g,c^2=g,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,e^b=ei,f^b=fi,h^b=hi,e^c=ei,f^c=fi,e^d=ei,h^d=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72282010
maximal normal subgroup of	#72282010
maximal quotient of	#72282185

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 310595);