#461101

order	512
nilpotency class	3
isoclinism	isoclinic class of #455570
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

893 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 368297);