#461154

order	512
nilpotency class	3
isoclinism	isoclinic class of #455828
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2631
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

894 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 368350);