#401682

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

1176 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)(11,19)(12,22)(13,25)(16,29)(18,28)(20,36)(23,24)(26,31)(27,32)(30,40)(34,37);(1,3)(2,5)(4,7)(6,8)(9,10,13,15,14,18,25,28)(11,16,21,26,19,31,35,29)(12,17,23,27,24,33,22,32)(20,30,36,38,34,40,37,39);(3,7)(5,8)(9,11,13,21,14,19,25,35)(10,16,15,26,18,31,28,29)(12,20,23,36,24,34,22,37)(17,30,27,38,33,40,32,39);(9,12)(10,17)(11,20)(13,23)(14,24)(15,27)(16,30)(18,33)(19,34)(21,36)(22,25)(26,38)(28,32)(29,39)(31,40)(35,37)

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,f^c=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, 308878);