#491417

order	512
nilpotency class	4
isoclinism	isoclinic class of #491245
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#121	D₈
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2966 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

387 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398613);