#404303

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	#62856	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2375 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)(11,20,23,15)(12,25,26,16)(13,18)(14,19)(17,27,35,30)(21,34)(22,31)(24,37,29,33)(28,36)(32,40)(38,39);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,22)(13,23)(14,29)(16,34)(17,32)(18,20)(19,37)(21,26)(24,28)(25,31)(27,39)(30,40)(33,36)(35,38);(2,5)(6,8)(9,12)(10,16)(11,21)(13,26)(14,27)(15,31)(17,19)(18,25)(20,34)(22,23)(24,39)(28,30)(29,40)(32,33)(35,36)(37,38);(1,3,4,7)(2,6,5,8)(9,12)(10,17)(11,22)(13,26)(14,30)(15,32)(16,19)(18,35)(20,38)(21,23)(24,39)(25,36)(27,28)(29,40)(31,33)(34,37)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 311499);