#489894

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	#3169	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

652 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,12)(4,15)(5,11)(6,10)(7,9)(8,13)(14,16)(18,24)(19,29)(20,22)(21,33)(25,41)(26,34)(27,39)(28,40)(30,32)(31,35)(36,37)(38,42)(43,45)(44,48);(1,3,7,4)(2,8,9,16)(5,14,6,13)(10,15,11,12)(17,18)(19,25)(20,26)(21,34)(22,27)(23,28)(24,33)(29,45)(30,42)(31,48)(32,44)(35,41)(36,40)(37,39)(38,47)(43,46);(1,4,7,3)(2,8,9,16)(5,13,6,14)(10,15,11,12)(17,19,23,32)(18,25,28,44)(20,29,22,30)(21,35,37,31)(24,38,40,43)(26,45,27,42)(33,47,36,46)(34,41,39,48);(2,9)(8,16)(10,11)(12,15)(17,20,23,22)(18,26,28,27)(19,30,32,29)(21,33,37,36)(24,39,40,34)(25,42,44,45)(31,47,35,46)(38,41,43,48)

Polycyclic presentation:

2^9|b^2=hi,c^2=h,d^2=i,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, 397090);