#404297

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	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2320 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)(11,20,16,12)(13,25)(14,18)(15,29)(17,27)(19,28)(21,37,33,40)(22,24,30,31)(23,38,36,39)(26,35)(32,34);(1,2)(3,6)(4,5)(7,8)(9,11)(10,12)(13,21)(14,16)(15,30)(17,23)(18,20)(19,31)(22,28)(24,29)(25,37)(26,33)(27,39)(32,38)(34,36)(35,40);(2,5)(6,8)(9,12)(10,16)(11,18)(13,23)(14,20)(15,31)(17,33)(19,22)(21,34)(24,28)(25,38)(26,36)(27,37)(29,30)(32,40)(35,39);(1,3,4,7)(2,6,5,8)(9,13)(10,17)(11,21)(12,23)(14,26)(15,32)(16,33)(18,34)(19,35)(20,36)(22,39)(24,37)(25,29)(27,28)(30,38)(31,40)

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,h^a=hi,c^b=cg,f^b=fi,h^b=hi,e^c=ei,h^c=hi,e^d=ei,f^d=fi,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, 311493);