#488538

order	512
nilpotency class	4
isoclinism	isoclinic class of #488531
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	C₂²×C₈	C₂⁴
prime radical	#60770	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#408 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

805 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 395734);