#414579

order	512
nilpotency class	3
isoclinism	isoclinic class of #411585
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	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1025 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:

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

Polycyclic presentation:

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

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, 321775);