#488619

order	512
nilpotency class	4
isoclinism	isoclinic class of #488531
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#12104	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#433 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

726 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|b^2=i,d^2=hi,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, 395815);