#490041

order	512
nilpotency class	4
isoclinism	isoclinic class of #489719
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	#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

667 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:

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

Polycyclic presentation:

2^9|a^2=h,b^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,d^b=di,e^b=eg,f^b=fh,g^b=gi,d^c=di,e^c=eh,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, 397237);