#404119

order	512
nilpotency class	3
isoclinism	isoclinic class of #394202
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	C₂×#2018	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1374 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)(11,20)(13,16)(18,28)(19,29)(21,36)(22,23)(24,25)(27,31)(30,39)(33,34)(35,37);(1,2)(3,6)(4,5)(7,8)(9,10,16,29,14,19,13,15)(11,17,20,31,23,32,22,27)(12,18,25,33,26,34,24,28)(21,30,35,40,37,39,36,38);(2,5)(6,8)(9,11)(10,17)(12,21)(13,22)(14,23)(15,27)(16,20)(18,30)(19,32)(24,36)(25,35)(26,37)(28,38)(29,31)(33,40)(34,39);(1,3,4,7)(2,6,5,8)(9,12)(10,18)(11,21)(13,24)(14,26)(15,28)(16,25)(17,30)(19,34)(20,35)(22,36)(23,37)(27,38)(29,33)(31,40)(32,39)

Polycyclic presentation:

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

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, 311315);