#491421

order	512
nilpotency class	4
isoclinism	isoclinic class of #491239
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	#19219	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2966 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

403 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)(18,23)(20,24)(21,31)(25,33)(26,34)(27,41)(29,37)(30,43)(32,38)(35,36)(39,48)(40,46)(42,47)(44,45);(1,3)(2,7)(4,13)(5,11)(6,12)(8,14)(9,15)(10,16)(17,18,24,36,21,27,32,34,22,26,38,41,31,35,20,23)(19,25,37,45,30,39,42,46,28,40,47,48,43,44,29,33);(1,3)(2,8)(4,11)(5,13)(6,12)(7,14)(9,16)(10,15)(17,19)(18,25)(20,29)(21,30)(22,28)(23,33)(24,37)(26,40)(27,39)(31,43)(32,42)(34,46)(35,44)(36,45)(38,47)(41,48);(1,4,6,5)(2,9,10,14)(3,11,12,13)(7,8,16,15)(18,26)(19,28)(23,34)(27,35)(29,42)(30,43)(36,41)(37,47)

Polycyclic presentation:

2^9|a^2=h,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=dhi,d^c=di,f^c=fh

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398617);