#499007

order	512
nilpotency class	4
isoclinism	isoclinic class of #498791
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#3061	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#41 ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1118 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321436, #72321850, #72321888
maximal normal subgroup of	#72321436, #72321850, #72321888
maximal quotient of	#72322628

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406203);