#404993

order	512
nilpotency class	3
isoclinism	isoclinic class of #394292
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	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1020 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,d^2=g,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,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, 312189);