#404258

order	512
nilpotency class	3
isoclinism	isoclinic class of #393921
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	#62845	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2379 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:

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

Polycyclic presentation:

2^9|a^2=ef,d^2=g,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei,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, 311454);