#489950

order	512
nilpotency class	4
isoclinism	isoclinic class of #489711
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#61477	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2966 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

703 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 72 points:

(2,6)(3,4)(7,8)(9,10,15,20)(11,21,25,49)(12,17,26,40)(13,29,33,47)(14,34,31,19)(16,36,39,23)(18,43,45,32)(22,48,50,70)(24,53,35,44)(27,56,54,69)(28,59,55,42)(30,46,61,38)(37,62,63,51)(41,66,65,58)(52,71,60,67)(57,68,72,64);(1,2)(3,6)(4,8)(5,7)(9,11,15,25)(10,16,20,39)(12,22,26,50)(13,30,33,61)(14,35,31,24)(17,37,40,63)(18,44,45,53)(19,46,34,38)(21,47,49,29)(23,43,36,32)(27,57,54,72)(28,60,55,52)(41,67,65,71)(42,68,59,64)(48,69,70,56)(51,66,62,58);(1,2)(3,6)(4,8)(5,7)(9,12,15,26)(10,17,20,40)(11,22,25,50)(13,27,33,54)(14,28,31,55)(16,37,39,63)(18,41,45,65)(19,42,34,59)(21,48,49,70)(23,51,36,62)(24,52,35,60)(29,56,47,69)(30,57,61,72)(32,58,43,66)(38,64,46,68)(44,67,53,71);(1,3,5,4)(2,6,7,8)(12,26)(17,40)(22,50)(27,54)(28,55)(37,63)(41,65)(42,59)(48,70)(51,62)(52,60)(56,69)(57,72)(58,66)(64,68)(67,71)

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 397146);