#483840

order	512
nilpotency class	4
isoclinism	isoclinic class of #483839
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#442
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂×#135	C₂×C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

438 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fg,e^2=g,f^2=h,g^2=i,b^a=be,c^a=cgh,e^a=eg,g^a=gi,c^b=ch,e^b=egi,f^b=fi,g^b=gi,d^c=di,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321998, #72322006, #72322075, …
maximal normal subgroup of	#72321998, #72322006, #72322075, …
maximal quotient of	#72343651

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391036);