#494292

order	512
nilpotency class	4
isoclinism	isoclinic class of #33522
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1100 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,d^2=i,e^2=hi,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,h^a=hi,d^b=di,e^b=eh,f^b=fh,h^b=hi,e^c=eh,f^c=fh,h^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321432, #72321844, #72321888
maximal normal subgroup of	#72321432, #72321844, #72321888
maximal quotient of	#72322829

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401488);