#401869

order	512
nilpotency class	3
isoclinism	isoclinic class of #395163
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
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

1202 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ei,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,h^c=hi,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, 309065);