#404115

order	512
nilpotency class	3
isoclinism	isoclinic class of #394292
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1334 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:

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

Polycyclic presentation:

2^9|b^2=e,d^2=g,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,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, 311311);